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Physics for Teachers:
A Modern Review
Physics for Teachers:
A Modern Review
Robert L. Weber
Associate Professor of Physics
The Pennsylvania State University
\ vV'
McGrawHill Book Company
New York Son Francisco Toronto London
to Marion, Robert, Karen,
Meredith, and Ruth
who were patient
»
PHYSICS FOR
WIQAN UBRARfES
WITH DRAWN FOR
EW
Copyright © 1964 by McGrawHill, Inc. All Rights Reserved. Printed in
the United States of America, This book, or parts thereof, may not be
reproduced in any form without permission of the publishers.
Library of Congress Catalog Card Number 6319320
68806
Preface
The reader of this boob is assumed to be interested in space
physics and atomic physics and to have had a fullyear course in
general physics at the college level. Capitalizing on his interest
in rockets and satellites. Part 1 presents enough of the principles
of mechanics and electricity to serve as a good basis for under
standing nineteenthcentury classical physics. In Part 2, with
the atom as the central theme, the theories of relativity and
quantum physics which have characterized twentiethcentury
modern physics are developed. In both parts important topics
such as wave properties and relativity, which are likely to be
less familiar to the reader, are developed in greater detail. Some
calculus notation is used, but where feasible (e.g., in Sec. 4.4),
noncalculus explanations are used, and several derivations in
volving integration are subordinated in an Appendix. An aim of
this book is to encourage the reader, whether a student or a
mature teacher, to appreciate the relatedness of the various
fields of science and to be willing to venture into new areas with
the ability he has gained from intensive study of a few selected
areas.
In the planning of this book I am indebted to the interest of
students and colleagues in several science institutes sponsored by
the National Science Foundation. For six years I served as
vii
Preface
associate director, director, and teacher of the physics part of
programs at The Pennsylvania State University; Colorado State
University, and Yale University. The present text evolved from
the study guide used in the lastmentioned program. I express
indebtedness to that scholarly textbook "Physics for Students
of Science and Engineering," by David Halliday and Robert
Resniek, for the manner of presentation used in the first part of
Chap. 13.
Contents
Robert L. Weber
Preface vll
Part 1 Looking Out: Rockets, Satellites, Space Travel 1
What's Up? 3
Environment of Space 9
Rocket Propulsion 20
Escape from Earth 40
Satellites 50
Motion of Bodies in Space 60
Travel to Moon and Planets 68
Part 2 Looking In: Atomic and Nuclear Physics 75
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
The Atomic Idea 77
Wave Motion 91
Electric and Magnetic Forces 116
The Electron 136
Ions and Isotopes 149
Electromagnetic Radiation
I Classical Theory 156
II Quantum Effects 167
Relativity Wonderland 184
Hydrogen Atom Bohr Model 203
Quantum Dynamics 216
Radioactivity 238
Nuclear Reactions 252
Absorption of Radiation 260
Unconventional Energy Sources 272
Appendices
A. Reaction Thrust 289
B. Burnout Velocity and Range 291
C. Schrodinger Wave Equation 293
D. 1. Books for a Physics Teacher's Reference Shelf 295
2. Some Periodicals for a School Science Library 296
3. Professional Organizations of Interest
to the Physics Teacher 296
4. Some Suppliers of Physics Apparatus for Teaching 297
5. Greek Alphabet 297
6. Symbols 298
7. Instruction on the Use of a Slide Rule 298
7A. Slide Rule Bibliography 300
8. Trigonometric Functions 301
9. Logarithms to the Base e 302
10. Values of Physical Constants 304
11. Conversion of Electrical Units 305
Index 307
PART
Looking Out:
Rockets, Satellites,
Space Travel
What's Up?
In his millennia of looking at the stars,
man has never found so exciting a
challenge as the year 1957 has sud
denly thrust upon him.
Fred L. Whipple
1.1 ASTRONAUTICS TODAY
Fictional accounts of space travel had been written before the
lime of Jules Verne. In the second century a.i>., Lucian of Santos
wrote of a visit to the moon. But the foundation for converting
fantasy into an engineering possibility was the invention of lite
rocket. Most current progress in the science and technology of
space flight is an outgrowth of the efforts since World War II to
develop longrange military missiles.
Popular concern aboul space technology was aroused when
the first artificial earth satellite was launched by the Soviet Union
in 1!J">7. In the next 3 years, some 36 satellites carrying instru
ments were launched, 30 of them by the United States. Also dur
ing this period several moon shots and space probes sought infor
mation farther from the earth. Next came the spectacular manned
orbital flights made l>y astronauts of the Soviet Union and the
United States.
3
4 Looking Out: Rockets, Satellites, Space Travel
In the "World Almanac" one may find a listing of the major
space vehicles launched since 19">7. Table 1.1 gives the longer
perapective of the history of man's use of rockets.
Table 1.1
c. 300
c. 1200
c.
1780
1792
c.
1800
1812
c
1830
1846
1913
C
1915
1926
c
1930
1931
1932
1936
1941
1942
1944
c
. 1945
1946
1949
c
. 1952
Milestones in missilery
Hero of Alexandria uses the reacting force of escaping steam to
propel an experimental device.
Chinese use gunpowder to propel "arrows of flying fire," equiva
lent to presentday skyrockets.
Advanced type of rocket developed in India.
Troops of Tipu, Sultan of Mysore, use rockets against British in
second Mysore War.
Sir William Congreve of Great Britain improves rocket propellant
to provide considerable increase in range.
British use rockets in attack on Fort McHenry (Baltimore), com
memorated in the line "... and the rockets' red glare" in our
National Anthem.
William Hale, an American, increases stability of rockets by adding
nozzle vanes.
Mexican War sees first use of rocket weapons by United States
in o war. Lifesaving rockets developed by English and German
inventors.
Ramjet proposed and patented in France.
World War I sees advent of guided missile to supplant aimed
rockets.
Dr. Robert H. Goddard, professor of physics at Clark University,
fires first successful liquid fuel rocket.
Germans experiment with the pulse jet, used to power the Nazi V1
"bun bomb" of World War II.
Germany uses liquid rocket fuel.
Captain Walter Dornberger undertakes development of liquid
fuel rocket weapons for the German Army.
German Peenemunde Project is organized, to develop war rockets.
United States starts work on controllable rocket weapons.
American Razon missile, controllable in both azimuth and range,
is developed.
United States government awards first contract for research and
development of guided missile to General Electric Company.
Germany uses V1 buzz bomb, V2, and other rocket missiles in
World War II. United States uses "Weary Willie" unmanned
bombers.
Work is started in the United States on an intercontinental ballistic
missile program, the MX774,
First flight of a missile beyond earth's atmosphere is made at
White Sands, N. Mex.
United States longrange missile program is stimulated by Atomic
Energy Commission warhead developments.
What's Up? 5
Table 1,1 Milestones in missilery (continued)
1954 United States starts ICBM program; USAF awards contracts to
Convair, North American Aviation, and General Electric.
1957 First artificial earth satellites — Sputniks I and II — launched by
rocket (October 4 and November 3).
1.31.58 Explorer I, first United States satellite, launched.
3.17.58 Vanguard I, first "permanent" satellite launched by the United
States.
10,11.58 Pioneer I, first lunar probe, launched by the United States.
12.18.58 Project Score (Atlas) launched, broadcasting a human voice from
outer space for the first time.
1.2.59 Russia launches Lunik, first satellite to orbit around sun.
3.3.59 Pioneer IV launched, first United States satellite to orbit sun.
9.12.59 Russia launches first space vehicle to land on moon.
10.4.59 Russia launches first satellite to orbit moon.
8.11.60 United Slates recovers first space vehicle from orbit.
4.12.61 Manned orbital flight achieved in Soviet Vostok satellite,
2.20.62 Pfoject Mercury succeeds in manned orbital flight,
7.10.62 Telstar satellite relays first transatlantic television, programs.
8.13.62 Two Russian astronauts put in reloted orbits.
8.27.62 Mariner II launched to encounter Venus.
12,14.62 Mariner II passed within 22,000 mi of Venus, reporting data on
temperature, cloud cover, magnetic field, particles and radiation
dosage encountered throughout voyage.
1.2 ASTRONAUTICS TOMORROW
What feat may be expected, perhaps in the next 2 or 3 years,
from adaptations of the intercontinental ballistic missiles already
available? The staff of the RAND Corporation has estimated
that we shall be able to do the following:
1. Orhit satellite payloads of 10,000 lb to 300 mi altitude
2. Orbit satellite payloads of 2,500 lb at 22,000 mi
3. Impact 3,000 lb on the moon
4. Land, intact, more than 1,000 lb of instruments on the moon,
Venus, or Mars
."). Probe the atmosphere of Jupiter with 1,000 lb of instruments
fi. Place a man, or men, in a satellite orbit around the earth for
recovery after a few days of flight
1.3 WHO SHOULD CARE?
By the military, the costly development of the rocket has been
pushed chiefly as a gunless artillery device and for bombardment
6 Looking Out: Rockets, Satellites, Space Travel
over intercontinental distances with nuclear warheads. Rocket
launched viewing satellites may make possible the inspection of
foreign territory and thus discourage preparation for war. The
difficulties that statesmen now have in reaching agreement on
inspection and on disarmament will, however, probably increase
as military space technology expands.
In addition to the military reasons, there are many scientific
incentives for making satellites. Some important problems await
ing investigation are:
1. Determination of density, pressure, and temperature in the
upper layers of the atmosphere
2. Exact measurement of the dimensions of the earth, the conti
nental distances, and other geodetic measurements
:i. A detailed study of radiation from the sun
4. Observation of the intensity of cosmic rays and other radia
tion in the earth's atmosphere
f). Correlation of the currents of nuclei, neutrons, and other
particles flying toward the earth with sunspot activity
(i. Kstimation of the distribution of mass in the earth's crust
from the orbital planes of the artificial earth satellite
7. Study of the propagation of radio waxes in the upper atmos
phere and provision of radio communication, navigation bea
cons, and television with the aid of satellites
8. Improvement in weather forecasting
it. Making feasible astro no mica I investigations without atmos
pheric and other disturbances
10. Study of biological specimens in environments different from
that on earth
Although we have these incentives for space exploration, it
is likely that such exploration will enlighten us in fields even
beyond our present speculations.
Space flight obviously demands development of devices of
great reliability 1o operate for long periods under extreme condi
tions of environment. Engineering advances depend on funda
mental scientific knowledge, and in the past these advances have
contributed tools for the obtaining of new knowledge. This inter
action or feedback is occurring in astronautics. When spaceships
can carry instruments, or man himself, into other parts of the
solar system, new information will surely become available for
the physicist, biologist, and astronomer.
What's Up? 7
The philosopher and the theologian are already adapting their
thinking to the eventuality that man may encounter life else
where in the universe. It seems probable that just as the tele
scope profoundly altered seventeenthcentury thought, the space
vehicle will extend twentiethcentury man's understanding of the
universe and his role in it.
1.4 CAREERS IN ASTRONAUTICS
Astronautics touches almost all fields of current science and tech
nology. 1 1 may be expected to lead to entirely new fields. Entrance
into the field of astronautics can be made by one who has acquired
knowledge in one or more of these fields: mechanical, aeronaut
ical, and electrical engineering; mathematics; physics; biophys
ics; and chemistry. Mathematics and physics are basic. With
nuclear power a necessity for distant space travel, the field of
nuclear physics is of special importance.
Information about careers in astronautics can be obtained
from the corporations active in this field, and also from the agency
which coordinates the government's activities, NASA. The .Na
tional Aeronautics and Space Administration was created by an
act of Congress signed by President Eisenhower on July 2H, I9">8.
The act declared that "it is the policy of the United States that
activities in space should be devoted to peaceful purposes for the
benefit of mankind."
FILMS ON ASTRONAUTICS
For a listing of some 9(1 films on rockets, missiles, and space science see
It. L Weber: Films for Students of Physics, Supplement I, American
Jcwntd of I'h units, SO: 321 327 (19«2).
SUGGESTIONS FOR FURTHER READING IN ASTRONAUTICS
Hooks:
Adams, Carsbie C: "Space Flight, " McGrawHill Book Company, Inc.,
New York, I95S, 373 pp.
Hcnson, t). 0., and H. Strugliold: "Physics and Medicine of the Atmos
phere ami Space," .John Wiley A Sons, Inc., New Yolk. IWiO,
•»4f) pp.
"uchheim. It. W„ and Staff of HAND Corp.: "Space Handbook,"
House Document 80, U.S. Government Priming Office, Washing
8 Looking Out: Rockets, Satellites, Space Travel
ton 25, D.C, 1959. Also Random House, hit. (Modern Library
edition), New York.
The National Aeronautic* and Space Administration: "Space: The
New Frontier," U.S. Government Printing Office, Washington 25,
D.C., 1962,48 pp.
Ordway, Frederick I.: "Annotated Bibliography of Spaee Science and
Technology," Arfbf Publications, P.O. Box 6285, Washing! on 15,
D.C., 10112.
Scifert, Howard S. (ed.): "Sparc Technology," .John Wiley & Sons, Int..
New York, 1959. Text based on graduatelevel lectures presented
by University Extension, University of California, in cooperation
with HamoWooldridgc Corp.
Periodicals:
Astronautics. Published monthly by The American Pocket Society,
Inc., 500 Fifth Ave, New York W. NY.
Aviation HVefc (Including Space Technology). McGrawHill Publishing
Company, Inc., 330 West 42 St., New York 36, N.Y.
Missiles and Rockets. Published weekly by American Aviation Publi
cations, 1001 Vermont Ave., NW, Washington 5, D.C.
Sky and Telescope. Published monthly by Sky Publishing Co., 60 Garden
St., Cambridge 38, Mass.
Space Aeronautics. Published monthly by ConoverMast Publications.
foe., 20.") East 42 St., New York 1 7, N.Y.
Space Age. Published quarterly bv Quinn Publishing Co., Kingston,
N.Y.
Spaceflight. Published bimonthly hy British Interplanetary Society,
12 Bessborough Gardens, London, SW 1. England.
Space Journal. Published quarterly by Space Enterprises, Inc., P. O.
Box 94, Nashville, Tenn.
Publications of Sperial Interest to Students:
Adams. Carsbie C, Wernher von Braun, and Frederick I. Ordway:
"Careers in Astronautics ami Rocketry," McGrawHill Book Com
pany, Inc., New York. 1902, 248 pp.
Map of the Moon, chart, 45 by 35 in.. General Electric Missile and
Space Vehicle Department, Valley Forge Space Technology Cen
ter (Mail: P.O. Box 8555, Philadelphia 1. Pa.).
Map of Outer Spaee, chart, 28 by 25 in., General Electric Missile and
Space Vehicle Department.
"Short Glossary of Space Terms," National Aeronautics and Spaee
Administration, U.S. Government Printing Office, Washington 25,
D.C, 1962, 57 pp.
"Space Primer: An Introduction to Astronautics," ConvairAsfro
nautics. Dcpl. 120, P.O. Box 112s, San Diego 12, Calif., 72 pp.
Environment
of Space
The most incomprehensible thing
about the universe is that it is compre
hensible. Albert Einstein
2.1 INTRODUCTION
Our sun, the 9 major planets, 31 known moons, and thousands
of lesser bodies all revolving around the sun constitute the solar
vii'iii. The planets move around the sun in the same direction
in elliptical orbits which are nearly circular (big. 2.1). All the
orbits lie in nearly the ecliptic plane of the sun's apparent path
among the stars. The orbit of Pluto deviates most, about 17°.
from the ecliptic plane. The zone about 17° wide on each side of
the ecliptic plane is known as the zodiac.
The average distance of the earth from the sun is 92,900,000
mi, a distance which is defined as one astronomical unit (a.u.).
The diameter of the orbit of Pluto, the outermost member of the
solar system, is about 79 a.u.
The four inner planets, Mercury, Venus, Earth, and Mars,
are sometimes called the terrestrial planets, They are relatively
9
10 Looking Out: Rockets, Satellites, Space Travel
small, dense bodies. The next four outer planets, Jupiter, Saturn,
Uranus, and Neptune, are called the major planets or the giant
planets. They are relatively large bodies with ice and rock cores
% >. Meteors
Jupiter
Fig. 2.1 Solar system; orbits of Mercury, Venus, Eorth, Mars, and
Jupiter.
below their visible atmospheres. Physical data on objects of prin
cipal interest in the solar system are given in Table 2.1.
Table 2.1 Physical data on some bodies in the solar system
Body
Mean
Man,
Diameter,
Gravita
Intensity
Length
Length
distance
times
mi
tional force
of
of day
of year
from
earth's
at solid
sunlight,
tun,
mass
surface.
rel. to
a.u.
9'«
earth
Sun
329,000
864,000

Mercury
0.39
0.05
3,100
0.3
6.7
68 d
88 d
Venus
0.72
0.82
7,500
0.91
1.9
?
225 d
Earth
1
1
7,920
1
1
24 hr
365 d
Mars
1.52
0.11
4,150
0.38
0.43
24.6 hr
1.9 yr
Jupiter
5.2
317
87,000
t
0.037
10 hr
12 yr
Saturn
9.5
95
71,500
t
0.01 1
10 hr
29 yr
Uranus
19.2
15
32,000
t
0.0027
11 hr
84 yr
Neptune
30
17
31,000
t
0.0011
16 hr
165 yr
Pluto
79
0.8
?

0.0006
?
248 yr
Moon
1.0
0.012
2,160
0.17
1
27 d
* Hoi no solid surface.
f location of solid surfoce not known (far below dense atmospheric gases).
2.2 THE SUN
The sun, whose gravitational attraction chiefly controls the mo
tion of planets in the solar system, is classified as about average
Environment of Space 11
among stars in size, in temperature, and in brightness (spectral
type C2). Its nearness to the earth makes the sun appear to us
very large and bright. The surface temperature of the sun has
Keen measured as about ti000°C, or I0,000°r', The energy output
of the sun as light and heat is remarkably constant. Solar energy
arrives at the surface of the earth at an average rate of 1.35 kilo
watts/ m. This solar energy, resulting from a series of thermo
nuclear reactions, makes life possible on the earth.
Sunlight takes a little more than 8 min to reach the earth.
When analyzed with a spectrograph, sunlight is found to consist
of a continuous spectrum, but with the colors crossed by many
dark lines. The absorption lines are produced by gaseous materials
in the atmosphere of the sun. From their lines, some 70 of the
chemical elements occurring on earth have been identified as
present in the sun.
The radiating surface of the sun is called the photosphere.
Above it is the chromosphere, visible to the unaided eye al times
of total eclipses as a turbulent pinkishviolet layer. The pearly
light of corona extends millions of miles beyond the chromo
sphere. Corona are related to the appearance of sunspots — dark,
irregular regions which may extend several hundred thousand
miles across and whicii may last for a few weeks to several months.
The output of ultraviolet radiation, radio waves ("static"),
and charged particles (cosmic rays) from the sun is highly
variable.
2.3 THE PLANETS
because of its nearness to the sun, Mercury is difficult to observe
ami knowledge of its physical characteristics is not very accurate.
Mercury has a mass about onetwentieth the mass of the earth. It
lias no moon. .Mercury has a rockysurl'ace. probably similarlof hat
»f our moon. Mercury always keeps the same side turned toward
the sun. This side probably has surface temperatures as high as
b)()°C, while the side in darkness is cold enough to retain frozen
gases, with temperatures approaching absolute zero.
Venus is slightly smaller than the earth, shrouded in a dense
; ilinospherc opaque to light of all wavelengths, Neither free oxy
Ren nor water vapor has been detected on Venus. Carbon dioxide
] s abundant in its atmosphere, with nitrogen and argon also
12 Looking Out: Rockets, Satellites, Space Travel
present. It is thought that die surface of Venus is hot {about
425°C), dry, and dark beneath a continuous dust storm.
Mars lias u diameter ahoul half that of the earth, its rate of
revolution is about the same as that of earth, and its axis, too, is
inclined about 2;">° from the plane of its orbit. Mars takes <»87 of
our days to make one circuit of the sun. Although the orbit is
nearly circular, it is not centered on the sun; Mars is more than
30 million mi farther from the sun at some parts of its year than
at others. More than half of the surface of Mars is a desert of
rusty rock, sand, and soil. The rest of the sinface shows seasonal
color changes which have been interpreted as due to vegetation.
While noon summer temperature on Mars probably reaches 30°C,
night temperatures probably fall to — 70°C. The atmosphere
{mostly nitrogen) on Mars has a pressure about 10 per cent of
the earth's atmosphere. Oxygen has not been detected. The white
polar caps are probably frost layers, which on melting furnish
moisture for the summer growth of vegetation. Mars has two
small satellites about 5 and 10 mi in diameter.
Each of the four giant planets, Jupiter, Saturn, Uranus, and
Neptune, seems to have a dense rocky core surrounded by a thick
layer of ice and covered by thousands of miles of compressed
hydrogen and helium with smaller quantities of methane and
ammonia. These planets receive relatively feeble radiation from
the sun, so that the temperatures of their upper atmospheres
range from 100 to 200°C. These planets rotate rapidly and
in the same direction. Some of the satellites of these planets are
larger than the earth's moon and may have physical character
istics less formidable for spaceflight visits than the major planets
themselves. Jupiter has 12 satellites. Four are bright enough to
be easily visible with binoculars, and their rapid motion causes
interesting changes in position from night to night.
Saturn is the farthest of the planets visible to the unaided eye.
It has nine satellites. Saturn is surrounded by remarkable flat
rings in the plane of the equator. It has been suggested that the
rings are made up of tiny particles of a shattered tenth satellite.
2.4 THE MOON
The moon has a mass about B V that of the earth, a diameter of
about 2,l(iO mi, and an elliptic orbit which gives it an average
Environment of Space 13
distance from the earth of 239,000 mi. The moon has no appre
ciable atmosphere. Its surface, comprising many craters and high
mountains, is probably dry, dustcovered rack. The moon rotates
on its axis in a period of time equal to the period of its revolution
about the earth, 27.3 days. The moon's elliptic orbit and its
variation in altitude from season to season permit us to examine
about 00 per cent of its surface, over a period of time.
2.5 ASTEROIDS, COMETS, AND METEORITES
Asteroids are pieces of planetlike material which, unlike the
planets, are of irregular shape. They may be the shattered frag
Fig. 2.2 Radiation pressure which
forces the tail of a comet away from
the sun might propel a spaceship.
* — V
meats of one or more planets. Most measure a few miles across;
the largest, Ceres, is nearly f>00 mi across. The orbits of most
asieroids lie between the orbits of Mars am! Jupiter {Fig. 2.1).
The time for one revolution varies greatly among the asteroids.
Comets are large, loose collections of material that penetrate
the inner regions of the solar system from outer space. The most
famous one, named after Halley, has been sighted every 7(i.02
years since 240 h.c, but not all return periodically. Comets have
a head and tail. The head is made up of heavy particles and is
attracted by the sun. The tail is made up of dust and gas and is
forced away from the sun by radiation pressure as the comet
sweeps past the sun. The brightness of the comets is probably due
to reflected sunlight. The earth has passed through the tails of
many comets without effect.
Some 2,000 tons of material from outer space reaches the
Ruth's atmosphere each day in the form of meteorite particles,
1'hose enter the earth's atmosphere with speeds of 10 to 50 mi/sec
and are heated to incandescence, producing the light streaks
called incteors. Reflection of radio waves from the ionized paths
14 Looking Out: Rockets, Satellites, Space Travel
and observations of sky gk)W at twilight as well as direct visual
counting of meteors indicate tliat a large amount of material
is received daily, but data are inadequate. How much meteoi
itie material a space vehicle might encounter is an important
unknown.
In the night sky a faint tapered band of zodiacal light can
„, ..en and traced photoelectrically. It is evidence of cosmic
dust, mierometeorites, concentrated toward the plane of the solar
system.
2.6 RADIATION
Beyond the shelter of the earth's atmosphere, x rays, ultravinl.t
rays, and cosmic rays exist at intensities which may have to be
considered in planning exploration by space vehicles. The WA
Explorer satellite detected an encircling belt of highenergy radMr
tion extending upward from a height of a few hundred miles, most
intense in the equatorial region (Fig. 2.3). The earth's magnetic
field traps the particles, chiefly electrons and protons, constituting
the radiation belt. During solar flares, the sun delivers as much
as 1,000 times its normal radiation.
The nature of the radiation found in space is described in
Table 2.2. In order to avoid subjecting astronaut* to radiation
Table 2.2 Radiation in space
Name
Nature of Charge Mass
radiation
Photon
Electronic
gnetic
Quantum
Electronic
gnetic
Xray
Electromagnetic
Gamma ray
Electromagnetic
Electron
Particle
— a
lm,
Proton
Particle
+■
1,840m,
or 1 omu
Neutron
Porticle
1,841m,
Alpha porticle
Particle
+ 2.
4 amu
Heovy primaries
Porticle
&+3e
' 6 amu
Where found
Radiation belts, solar radia
tion (produced by nuclear
reactions and by stopping
electrons)
Radiation belt
Cosmic rays, inner rodiotion
belts, solar cosmic rays
Vicinity of planets and sun
(produced in nuclear inter*
actions — decoys into pro
ton and electron)
Cosmic roys (nucleus of
helium atom)
Cosmic rays (nuclei of heav
ier otomsl
From H. E. Newell and J. E Naugle, Science, 132i 1*65 (1960).
Fig. 2.3 Space radiation: cross section of the radiation pattern in longitude
75° west, from Explorer satellites. (Adapted from the New York Times,
October 27, 1958.)
'9 2,4 Pioneer V, o 951 b highly instrumented space planetoid, was launched
March 11, 1960, to supply the first comprehensive data collected in interplane
,ar y space.
15
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Environment of Space 17
in excess of tolerable dosages (Table 2.3), it may be necessary
to plan flights from the earth along trajectories which avoid the
regions of concentrated radiation (Fig, 2,">).
Table 2.3 Maximum permissible radiation dosages and some typical
exposure levels {in roentgens)
Item
Amount
Permissib/e exposures
Maximum permissible dosages
Maximum permissible emergency exposure
Typical exposures
Normal radiation level (sea level)
Undisturbed interplanetary space (cosmic rays)
Heart of inner belt (protons)
Heart of outer belt (soft x rays)
So!ar proton event (protons)
Total exposure
0.3* r/'quarter
5.0 r/yr
25 r
0.001 r day
512 r/yr
24 r/hr
~200 r/hr
1010" r/hr
2400 r
E Ei
* Limit prescribed for radiation workers. Under this limit the yearly
maximum would be 1.2 r.
From H. E. Newell and J. E. Naugle, Science, 132: 1465 (1960).
2.7 MORE DISTANT SPACE
The sun's nearest star neighbor is Alpha Centauri, which is more
than 4 lightyears away. (In contrast, the outermost planet of
the solar system, Pluto, is only 0.000 lightyear from the sun.)
The relative brightness of several stars and their distances from
the sun are shown in Table 2.4.
Insofar as man knows, the universe is infinite. Scattered
throughout this void is an apparently endless number of galaxies,
each of which contains millions of stars. Some galaxies are them
selves grouped in clusters. The constellation Corona Borealis is
made up of some 400 galaxies. In the observable region around
us there are an estimated I billion galaxies, with an average dis
tance between galaxies of about 2 million liglil years.
Galaxies usually have the shape of disks thousands of light
years in diameter. The larger galaxies have spiral arms suggesting
a pattern of rotation. Our own galaxy, the Milky Way, appears
tn have this form. Our solar system is believed to be situated in
16
18 Looking Out: Rockets, Satellites, Space Travel
Table 2.4 Some star distances
Star
Brightness,
relative to sun
Distance from sun,
lightyeors
Alpha Centauri
Barnard's Star
Wolf 539
Sirius A,B
Proeyon A
Altair
Argo
Deneb
Betelgeuse
1
1 60,000
23
6
8
5,200
6,600
13,000
4.3
6.0
7.7
8.7
11.3
16.5
180
640
650
one of the spiral arm*, about 30,000 lightyears from the center.
The solar system is moving at a speed of about 1.0 mi see, but
it takes 200 million vears to complete one circuit of the, galaxy.
Ylthough presently envisioned techniques may lead to
manned exploration in the solar system, they will not suffice for
exploring (he vast distances beyond.
If as has been estimated, less than 12 per cent of all stars
have planetary systems, then nut of some 200 billion stars m our
galaxy, there are some billion with planetary systems. One is led
to speculate that out of this number there are probably some
systems with earthlike planets that may support life. Communi
cation with distant planets of the galaxy is a matter «,t specula
tion only. And bevond our galaxy arc other galaxies at least out
to the limits accessible to present telescopes: some '2 billion light
years in all directions.
A FILM INTRODUCTION TO SPACE
Ut*mB.28wto (I960), National Fiim^r,Ur Ca^la. For vent from
Contemporary Films. Inc., 287 West 25 St, New York I, N.Y.
Teachers' guide available,
SUGGESTIONS FOR FURTHER READING
Baker, Robert H.: "Astronomy," 7th e<L, I). Van Nostrum! Company,
Inc., Princeton, NJ. t 1989. „ „.,
Bauer Carl \ : "The Universe bom fee Known to the Unknown, n»
Pennsylvania Stale University. Continuing Kdueation, Umvcrsitj
Environment of Space 19
Park, Pa., 1982, 54 pp. A manual for adult discussion study groups
to which a Guide for the Discussion Leader is keyed,
Duncan, John C: ■■Astronomy," 5th ed., Harper it' Row, Publishers,
Inc.. Now York, 1055.
Hoyie, Fred: "Frontiers of Astronomy." New American Library of
Winid Literature, Inc., New York. 1955.
James. .1. N.: The Voyage of Mariner II, Scientific American, 209:
7084 (1983).
Krogdfthl, Wasley S.: "The Astronoinical [.'inverse," The Macmillan
Company, New York, 1952.
McLaughlin, Dean II.: "Introduction to Astronomy," Houghton
Mifflin Company, Boston, 1981.
Newell, II. !■",.. and J. K, Xaugle: Uadiulion Environment in Space.
Sriri„r. 1:12:1 if 15 I 172 (Mil ill).
Shaw, John II.: The Radiation Environment of Interplanetarv Space,
Applied Optics, I . 8795 (1902).
Zim. Herbert, and Robert 1L linker: "Stars." Colder, Press. Xew York
1950.
QUESTIONS FOR DISCUSSION
1. Would you expect Mercury to have an atmosphere, that is, a
permanent gaseous envelope?
2. Estimate the fraction of the total mass of the solar system which
» i" I lie sun. Am. (iO.fi per cent
3. Does a physical environment of the sort needed to support plant
and animal life such as we know exist elsewhere in the universe? Where?
Can you conceive of a form of life not based on water chemistry?
Might ammonia or fluorine compounds serve? Where in the universe
would you expect lids differenl form of life to exist? If it docs exist in
intelligent. r,, rmi ,[,, vnu tM j n k Wl , ( . ou j (  ,. tHnmu „i t . at p ,vilh it?
I. A point on the earth's equnlor is carried ahoul 1,090 uii.hr by
the rotation or the earth. Jupiter has an equatorial diameter II times
thai of the earth and a day of 10 hr. Calculate (he speed of a point on
the equator of Jupiter. .W _>2.i mi hr
5. Express the diameter of Pluto's orbil in mi.
Ami. 7,309 million mi
6. In what ways is it true that all our sources of energy— plant life,
"oal. oil, and water arc derived from the radiant energy we receive
uom the sun?
T. Furnish some evidence for or against the statements: The climate
"ii Mars is similar to that which one would encounter on a iOmihigh
' ' 'i on earth. Granted transportation, a selfsustaining colon v might
'"' established on Mars.
8. The four outermost of Jupiter's 12 satellites revolve about
• npitei rrom east to west, contrary to Ihe motion of most satellites in
i solar system and to the direction of revolution of Ihe planets around
1 «m. Can yon suggest a possible reason for this retrograde motion?
Rocket Propulsion 21
Rocket
Propulsion
Necessity is not the mother of inven
tion; knowledge and experiment are
its parents. W. R. Whitney
All vehicles move !>y reaction with some other thing. Cars require
traction on the road. Snip* and planes push or pull themselves
through water or air. Only rockets carry their own "something
else" to push against. In the words of J. X. Savage, "a rocket is
any machine that propels itself by ejecting material brought
along for the purpose."
A rocket is an internalcombustion engine that carries its own
supply of oxygen (in any of several forms of "oxidizer") There
fore, it does not require air but can operate in a vacuum, as in
space.
The description of a rocket in flight is a particular application
of the general theory of the dynamics of rigid bodies. It is con
venient to consider separately the motion of the center of mass
and the motion of the body around its center of mass. The former
is the theory of flight performance, the latter, the theory of sta
bility and control. The powered flight of a ballistic rocket is usu
20
ally, for practical reasons, confined to two dimensions. So the
theory of motion in one plane is adequate. In further simplifi
cation, we may begin by considering the flight path to be a
straight line (onedimensional theory).
We shall consider in this chapter the basic principles of rocket
propulsion; the effects of mass ratio, specific impulse, and 1 1 mist/
weight ratio on the flight of a singlestage rocket; and then the
performance of multistage rockets.
3,1 MECHANICAL PRINCIPLES: NEWTON'S LAWS
Three laws formulated by Sir Isaac Newton in the seventeenth
century are fundamental to rocketry:
1. A body at rest remains at rest and a body in motion continues
to move at constant speed in a straight line unless acted upon
by an externa}, unbalanced force.
2. An unbalanced force acting on a body produces an acceleration
in the direction of the net force, an acceleration that is directly
proportional to the force and inversely proportional to the
mass of the body.
3. For every force that acts on one body, there is a force equal
in magnitude but opposite in direction that reacts upon a
second hody.
Table 3.1 Consistent systems of units for Newton's second law
Name of system
Unit of mass
Unit of force
Unit of
acceleration
Mks (absolute)
kilogram
newton*
meter, second 
Cgs absolute
gram
dyne*
centimeter second 
Cgs gravitational
No name assigned
m = W, g
gram
centimeter, second 3
British absolute
pound
poundal*
foot/second 2
British gravitational
slug*
pound
foot /second 1
Any system
W, g
Same unit as that
Same unit as that
used for
W
used for g
In each set the starred unit is the one usually defined from the second law
so as to make k = 1 in F = kma.
22 Looking Out : Rockets, Satellites, Space Travel
3.2 UNIFORMLY ACCELERATED MOTION
It is convenient to list and remember the equations which apply
to a body which moves with constant acceleration in a straight
line. This is a special case, but one often met. The average speed
v is the distance traveled divided by the time required, 5 = s/t, or
, = U M>
Since we have assumed motion in a constant direction, the accel
eration is the change in speed divided by the time, a  (»i — Vi)/t,
or
v,  „ = at 03)
Since the speed changes at n uniform rate, the average speed f is
equal to half the sum of the initial and final speeds:
»! + "« (3.3)
2
By combining these, two other useful equations can be obtained.
Eliminating u s and v, we get
s  M + Jo* 1 < 3  4 >
By eliminating 6 and t from Eq. (3,1) to (8.3), we get
S =
3.3 REACTION PRINCIPLE
A rocket engine develops thrust by employing Xewton's third
law in the following manner. Imagine a stationary sphere (Fig.
3.1«) containing a combustible mixture of gasoline vapor and air.
p F
(a) (6) M
Fig. 3.1 Reaction, the principle of jet propulsion.
a Ah P,
Rocket Propulsion 23
If this mixture were ignited, there would result a high pressure
p t in the chamber exerting force equally in all directions. The
sphere would remain at rest because there would be no net force
acting on it.
Consider a section to be removed from one side of the sphere
so that the gases could escape. The sphere would now experience
a net force. Since there would be no balancing force across area
A\ (Fig. 3.1b), the force on area A> would cause the sphere to
move to the left. The magnitude of this force or thrust F would
be equal to the product of the pressure p c in the chamber and the
area A t of the throat:
F = pAt
cm
A greater force can be obtained under certain conditions (Sec.
3.(>) by adding an expansion nozzle at the exit (Fig. 3.1c). The
contribution of the nozzle is represented by a thrust coefficient
Cp used as a multiplier in the previous thrust equation, so
F = PrAtCp
(3.7)
From Xewton's laws, if F is the net external force applied to
a system, the rate of change of momentum of the system is
A(mi>) _ „
(3.8)
When a rocket is in free space, the net external force acting upon
the rocket is zero. If mass particles are ejected from the rocket
with a constant exhaust velocity iv, their rate of change of momen
tum gives the rocket an accelerating force
'  " A, *
(3.9)
The negative sign expresses the fact that F and c, are in opposite
directions.
The exit pressure p„ of the gas from a rocket often may be
either greater or lower than pressure of the racket's environment,
ambient pressure p«. Also, while p r remains constant, p„ will de
crease as the missile gains altitude. If the difference between the
two pressures is multiplied by the exit area .4,.. we have the mag
nitude of the unbalanced force (p c — p a )A e acting on the rocket.
24 Looking Out: Rockets, Satellites, Space Travel
This force is called the "pressure thrust," in contrast with the
"momentum thrust" expressed in Eq. (8.9).
The total thrust of a rocket engine can be expressed as the
sum of the momentum thrust and the pressure thrust:
F =  ^T V* + (}Je  V) A 
(3.10)
"Usually, the only term on the righthand side of this equation
that will vary with respect to time is p„, the ambient pressure.
66
64
/"
. 62
S
o
f
X
X
=e 60
/
1
/
£ 58
 /
56
54
i il _!
20,000 40,000 60,000
Altitude, ft
80,000
100,000
Fig. 3.2 Decrease of thrust with altitude, V2 missile.
Positive thrust Negative thrust
t
Fig. 3.3 Pressure thrust in a rocket motor.
The resulting decrease in thrust as a missile gains altitude is
shown in Fig. 3.2. If, as often at sea level, p e < P, the pressure
thrust term will be negative (Fig. 3.3).
Rocket Propulsion 25
3.4 EFFECTIVE EXHAUST VELOCITY
In order to simplify the thrust equation, an effective exhaust
velocity >\. fi is defined as
F
V,n Am A!
Then Eq. (3.10) is written in simplified form as
(3.11)
_ Am
(3.11a)
Of course v M is variable with altitude, whereas », is constant for
a particular rocket system. Cnder optimum conditions for expan
sion of the gas, when p, = p„, the effective exhaust velocity e,. f r
if equal to the theoretical exhaust velocity r,.
3.5 SPECIFIC IMPULSE
The performance of a rocket engine is conveniently described by
its specific impulse. This is the thrust produced divided by the
weight of propellant consumed per second
Aw/At
(3.12)
Since F and w are expressible in the same unit (e.g., pounds), the
unit for specific impulse is the second. If other factors are held
constant, the speed that a missile can attain is directly propor
tional to the specific impulse of its propellants.
The specific impulse varies with altitude, since thrust is vari
able with altitude. By combining Kqs. (3.11) and (3.12), the
following useful relationship is obtained:
j _ /■' _ V v it Am / At r,. fi
Air/ At " i) Am II ' ~f
(3.13)
To avoid the difficulty of having /, become infinitely large as the
gravitational acceleration g approaches zero at high altitudes, it
is generally agreed that the value g u = 32.2 ft/sec si mil be used
in K<). (3.13):
i t — —
(314)
26 Looking Out: Rockets, Satellites, Space Travel
The simplified graph (Kig. 8,4) b intended to summarize the
facts that certain quantities, such as theoretical exhaust velocity
u„ propcllant flow rate Am /At, gas pressure in chamber p„ and
exit pressure p, are constant for the rocket system. Other quanti
ties, such as thrust F, effective exhaust velocity v Mt and specific
F> "•«' 4
*£.v e ,P e ,P c
Sea level
Altitude
150,000 ft
Fig. 3.4 Simplified representation of the fact that dm, dt, v„ p„ and
p t are independent of a rocket altitude, while F, v,ti, and (, increase
with altitude since they depend on ambient pressure p a .
impulse /„, vary with altitude since they depend upon the ambient
pressure p a .
Table 3.2 Performance of typical liquid propellant combinations (calculated
for expansion from 300 lb in. 1 to 1 otm)
Propellant combination
Mixture Exhaust Specific
ratio velocity, impulse,
(oxidizer/fuel) ft/sec sec
Liquid oxygen and 75% ethyl
alcohol,
25% H 5 (V2 propellant)
1.3
7700
239
Liquid oxygen and liquid H,.
5.33
10,800
335
Liquid oxygen and kerosene
2,2
7,970
248
Fluorine and hydrazine
1.9
9,610
299
H.0;:(S7%) and H,.0 (13%)
4,060
126
Red fuming nitric acid and aniline
3.0
7,090
221
Ni from ethane
7,010
218
From G, P. Sutton, "Rocket Propulsion Elements," John Wiley &. Sons, Inc.,
New York, 1949.
Rocket Propulsion 27
As the measure of overall engine performance, specific impulse
is related to both combustion performance and expansion per
formance. From thermodynamics il may be shown that
<£
(3.15)
where T is the combustion temperature and M is the molecular
mass of the exhaust gas. Thus a hot, lightweight gas gives a high
specific impulse. Roth a large value for the heat of propellant
combustion and low specific heat of the gas are desirable to pro
duce the high temperature. The requirement of lowmolecular
mass suggests that the products of combustion should be rich
in hydrogen compounds.
3.6 FUEL COMBUSTION AND EXPANSION
The basic principles we have been discussing and some refine
ments in design can be illustrated by considering a typical rocket
engine using a liquid fuel and oxidizer (Kig. 3.5). The engine con
Combustion De Laval
chamber nozzle
Fig, 3.5 Simplified liquid racket motor.
verts the thermochemical potential energy of the propellants into
the kinetic energy of the gas in the exhaust jet. The steps involved
;nc propellant feed, injection, ignition, combustion, and expan
sion. Tin: liquid propellants arc forced from their tanks into the
injector by means of compressed gas or a turbopump. The injector
distributes the fuel and oxidizer in a flow pattern that causes
thorough mixing. Ignition is started by a device at the surface
of the injector; thereafter heat from the combustion gases main
28 Looking Out: Rockets, Satellites, Space Travel
tains continuous ignition. Combustion takes place throughout the
combustion chamber with some residual burning in the exhaust
gas jet.
During combustion, the propellants change from liquid to
gas, and by electron sharing they combine to make new chemical
compounds. Chemical potential energy is converted into thermal
energy, raising the gas to a very high temperature. The change
from the liquid to the gas state plus the high temperature of the
gas results in a high chamber pressure. Gas particles are forced
to the rear.
It is the purpose of the nozzle to allow the gases to leave the
rocket in smooth flow and also to accelerate these gases. The rear
of the combustion chamber first converges to a throat area A t
and then expands to an exit area A m which may have about the
same diameter as the combustion chamber. The change from
potential energy (nondireeted thermal motion of gas atoms) to
the kinetic energy of a highvelocity gas jet occurs in two steps.
As the gas passes through the converging portion of the nozzle,
the decreasing crosssectional area causes the flow to speed up.
The gas flow reaches a maximum speed corresponding to sonic
speed (Mach 1)* at the nozzle throat provided the chamber
pressure exceeds a critical value, approximately twice the sur
rounding atmospheric pressure. The addition of a diverging nozzle
provides for even more acceleration of the gases. A typical throat
speed may be 4,000 ft/sec and exit speed 7,000 ft/sec. The expan
sion area rat in
A,
A t
Cilo)
is chosen for a particular engine to give the highest average
thrust over the powered portion of the trajectory, For a given c
a bell nozzle may be some 30 per cent shorter than a conical
nozzle, and hence its use conserves rocket weight.
An interesting phenomenon called jet separation may add
additional thrust. When the exit pressure is very low in compari
son with the ambient pressure, gas flow breaks away from the
* Mach number M is defined as the ratio of Free stream speed v to
the local speed of sound a, M = v/a\ it is the ratio of directed molecular
motion to random molecular motion.
Rocket Propulsion 29
wall before reaching the nozzle exit. The thrust coedieient is
slightly higher during separation than for a fullflowing nozzle.
Jet separation .
Optimum
expansion
Jet separation
Under
expansion
Pe > Pa '
expansion
Pe <Pa
Expansion area ratio, 6
Fig. 3.6 Jet separation.
3.7 BURNOUT VELOCITY AND RANGE
Consider the case of a rocket moving in a straight line inclined at
an angle 8 with respect to the direction of gravity, with thrust F
paraDel to the path. The equation of motion will be
to
m
At
— F — D + mg cos
(3.17)
where D is the aerodynamic drag and g is the acceleration of
gravity at the location of the rocket. Since D usually depends on
Fig. 3,7 Forces on a rocket,
the shape and speed of the rocket and the density of the snrround
n»g air, let us assume for this illustration that the rocket is at such
30 Looking Out: Rockets, Satellites, Space Travel
high altitude that D = 0. If we divide Eq. (3.17) by m and use
Eq. (3.13) to set F = gJ.Am/At, we have
1 Am
t'^'irs
(3.18)
If we assume tliat the rocket starts from rest, v = 0. We set
/« = tVi/ff and let R he the ratio of initial mass to final burnout
mass, R = m u /m b . Then E(. (3.18) can be solved (Appendix »)
to find the velocity a at burnout
vt = hu In R — gh cos (3.19)
where k is the duration of burning. The two averages iv« and g
are necessary since the values of both effective exhaust velocity
and gravitational acceleration are dependent on altitude.
Greater range and less time for interception of a rocket will
result from increasing the burnout velocity of the missile. This
improvement, can he obtained, according to Eq. (3.19), by
increasing the effective exhaust velocity and the mass ratio.
3.8 MASS RATIO
The mass ratio is defined as the quotient of the initial or total
mass m of a rocket and its burnout mass »h 4 :
III:,
(3.20)
This is one or several dimension less ratios useful in comparing
rocket designs. Others, whose definitions should lie apparent, arc
the thrust, fullweight ratio, the deadweight fraction, and the
payload fraction. The burnout mass is related to the initial mass
simply by
(3.21)
Am ,
where Am /At is the propellant flow rate,
From Eqs. (3.18) and (3.19) it is evident that to achieve the
desirable high burnout velocity, a fuel with high specific impulse
is needed. Further, for a given value of /„ larger mass ratios pro
vide higher values of iv The mass ratio of the World War II
German V2 rockets was about 3.2. For present rockets R U
as high as .">.
Rocket Propulsion 31
3.9 MULTIPLESTAGE ROCKETS
In a singlestage rocket the propulsion energy must be used to
accelerate the entire empty mass of the rocket even after most of
that empty mass is no longer useful. This severely limits the
velocity attainable. Tn fact, with present fuels, a singlestage
rocket cannot achieve velocities of the order of 25,000 ft/sec and
higher required to place a satellite in orbit or to escape the earth's
gravitational field.
A multiplestage rocket is made up of a number of independ
ent sections each equipped with a propulsion system and a portion
of the total propellant load. After the first (booster) stage has
lifted the entire rocket and has reached its burnout velocity, its
empty mass is dropped from the rocket. A second (sustainer)
stage carrying the payload is then fired and continues to accel
erate the now lightened missile to the appropriate final velocity.
( )f course more than two stages can be used, but design and oper
ational difficulties become more numerous as stages are added.
If each of a series of stages has the same values of specific impulse,
deadweight fraction, payload fraction, and thrust/ weight frac
tion, each will contribute the same increase in velocity to the
final payload. This design results in the lightest overall rocket
to perform a given mission.
A simplified expression for the burnout velocity of a twostage
rocket is
Vt = fvrln (/r,/rs) (3.22)
Here R t is the initial mass of entire rocket divided by the burnout
mass of the first stage plus the initial mass of the second stage,
R = (wioi + v»us)/("'m + »t 02) and R.< m hi,,., ,jj,,,. if the second
stage is made small in relation to the first stage, the value of the
logarithmic term in Eq. (3.22) will he greater than that in Eq.
(3.19), predicting a greater final burnout velocity for the two
stage rocket than that given by Eq. (3.19) for a singlestage
rocket in vertical flight, namely, 1% = i\.rr In R.
310 NUCLEAR PROPULSION
Some advanced concepts for rocketpropulsion systems have to
do with development of recoverable boosters, restartable engines,
s1 "nit>k' propcilants. and nozzles which allow a reduction in
32 Looking Out: Rockets. Satellites, Space Travel
engine size. But efforts are also being made to find new sources of
rocket power, other than chemical reactions;. Figuratively, we
should like to be able to pack the power output of Hoover Dam
(1.3 X 10 6 kilowatts) into a sports car. The development of
nuclear power sources promises to provide specific impulses sig
nificantly greater than the values, around 400 sec, for chemical
fuels.
Research on the use of a nuclear reactor as a rocket energy
source has been carried out since 1955 in Project Rover, directed
by the Atomic Energy Commission and the National Aeronautics
Pressure shell
Nozzle
Reactor core
Fig. 3.8 Scheme for a nuclearpowered rocket engine.
and Space Administration. The test engines have been named
Kiwi's, after a flightless bird. Heat is generated in solidfuel
elements by nuclear fission (('hap. 20). Hydrogen gas flows
through channels in the core. The heated gas is exhausted at high
speed through a nozzle (Fig 3.8).
The thermodynamic (Caniot) efficiency of any heat engine
is given by
7', ■ T>
Ti
where 7'i is the temperature (absolute) of the source of energy
and Ta is the temperature at which the working fluid is dis
charged. The lieat exchanger nuclear engine exhausts into a
relatively low temperature environment, especially when in
Efficiency =
(3.23)
Rocket Propulsion 33
Fig. 3.9 KiwIA nuclear engine at Project Rover test site in Nevada.
space. So one would expect to be able to put almost all the
nuclear energy into thrust. The limiting factor is the energy den
sity one can put into the propellant to eject it at sufficiently
high speed.
\ssumc thai one has an ideal nozzle to recover directed
kinetic energy from the thermal motion of the propellant mole
cules and that the propellant acts as a perfect gas. Then
W
RT
(3.24)
and the exhaust velocity v c is proportional to \/T/p or to \/T/M,
where p is the density of the propellant, M is its molecular mass,
and Ft is the universal gas constant. For high velocities one wants
maximum temperature and minimum molecular mass. Thus
hydrogen heated to the highest feasible temperature gives the
largest specific impulse of any material. Estimates range from
,J 00 to 1,500 sec for the specific impulse of a heat ex changer
34 Looking Out: Rockets, Satellites, Space Travel
nuclear rocket engine. Recalling the relation n, = 7» p ln (mo/m,),
it is apparent that the larger l. p attainable with nuclear pro
pulsion allows one to reach a desired orbital velocity or escape
velocity with a much lower initial fully fueled weight (smaller
mass ratio )«<>/'«»)•
Under Project Sherwood, studies are being conducted to find
ways of controlling and using the energy liberated in the fusing of
the lightest nuclei into heavier nuclei. The phenomena involved
in thermonuclear (fusion) rockets, plasma rockets, and ion rock
ets fall under the general term magn^ohydrodynamica: the study
of the behavior of ionized gases acted upon by electric and mag
netic fields.
Deuterium is a likely fuel for a fusion rocket. Heated to a very
high temperature, the deuterium would maintain a highspeed
plasma (hot, ionized gas) capable of specific impulses rated in
millions of seconds. There is a difficult problem in confining a
plasma at the temperatures estimated to he around :i">0 million
degrees. Perhaps the plasma could be kept from coming into con
tact with material walls in chamber and nozzle by suitably shaped
magnetic fields.
3.11 ION PROPULSION
The removal of one or more electrons from molecules of a propel
laut, by passing the propelknt through heated metal grids, pro
vides ions which can then be accelerated to high velocities through
a nozzle by an electric field. Volt age takes the place of tempera
ture in producing acceleration.
One such technique uses metallic rubidium or cesium prope.l
lant and tungsten grids. Each time an atom of cesium comes in
contact with the heated tungsten grids, an electron leaves the
cesium atom and goes to the tungsten metal. The resulting cesium
ions travel past decreasing potential levels and arc accelerated
to their final exhaust velocity.
The ion rocket will always have relatively small thrust. It
will require assistance (from chemical or nuclear rockets) in
ground takeolTs where strong gravitational force must be over
come. But the performance of an ion engine at high altitudes will
be very good. Estimates of its specific impulse are as large as
20,000 sec. The amount of electric power required for an ion
Rocket Propulsion 35
rocket is very large. The weight of the electric power plant, even
using nuclear fission or solar radiation devices, is a major obstacle
to an efficient ion rocket.
Electrons
Distributor plots and
housing ot 40,000 volts >.
s El eel
Va— i
Electric generator
f Electrons
Propel I ont
Heater coils ^Ionization gr
20,000 volts''
10,000 volts
5,000 volts
Fig. 3.10 Scheme for an ion rocket engine.
Propellent injection
Arc discharge
High current
circuit
Fig. 3.11 Scheme for plasma rocket engine.
3.12 PLASMA PROPULSION
Hie propcllant may be heated directly by maintaining a powerful
electric arc in it. In this way high temperatures can be obtained,
leading to a specific impulse of perhaps 2,000 sec. But this device,
36 Looking Out : Rockets, Satellites, Space Travel
too, will require 8 great amount of electric power, about (50 kilo
watts for each pound of thrust.
In plasma and ton propulsion the thrust can be applied con
tinuously over an extended period of time. Hence by these tech
niques one can propel in space rockets whose weight on earth
greatly exceeds the thrust of ion propulsion.
3.13 SOLAR PROPULSION
In one scheme of solar propulsion, the radiation pressure of solar
rays (ailing on a "sail," perhaps a lightweight reflecting sphere,
attached to the spaceship would propel if. In another scheme, the
Sun's rays heat water
circulating
at Focus of mirror,
producing steam
Steamdriven
turbogenerator
to develop
Solar sail
Fig. 3.12 Schemes for solar propulsion, (o) Steam generated by
solar energy drives electric generator. {b\ Recoiling photons
impart momentum to sail.
Rocket Propulsion 37
solar rays would he used to heat hydrogen gas which would then
lie expelled through a nozzle. Kstimated values of the thrust are low
hut are several hundred times those of an ion or a fusion system.
3.14 MODEL ROCKETRY
Many a youth has felt the urge to become a backyard rocketeer.
The National Association of Rocketry, founded in 15)57, seeks to
advance model rocketry as a scientific hobby and as ati edu
cational program. The NAR has developed rules and procedures
for a safe, supervised, citizenoperated modelrocketry program
for enthusiasts of all ages.
Model rocketry is concerned with small, light, inexpensive
rockets made of paper, balsa, plastic, and other noninetallic
materials, powered by commercially available rocket motors.
Emphasis is placed upon design, performance, flight character
istics, instrumentation, and reliability. Competitions are spon
sored hy local societies.
I'iiblicat ions ami informal ion about the XAR can lie obtained
from G. Harry Stine, President, National Association of Hock
el rv, Stamford Museum it Nature ( "enter, Stamford. Connecticut.
Physics teachers may he interested in model rocketry as a
device for stimulating student interest in mathematics, mechan
ics, aerodynamics, meteorology, electronics, optics, and pho
tography. Ideas based on the experience of the most active sections
of the NAIl may be requested from Dr. Stine at the address
above.
MANUFACTURERS OF MODEL ROCKETS
American Telasco Limited, 135 New York Ave., IhUesitc, X.V.
Centuri Knpncering Co.. 340 \V. Wilshirc Drive, Phoenix 3, Ariz.
Ci>a»ler Corporation, P.O. Box 2S0, Hiiless. Tex.
Bates Industries, Inc.. P.O. Box 227. Penrose, Colo.
Model .Missiles, Inc., 2ti!<) Bast Cedar Ave., Denver 22, Colo.
Propulsion Dynamics. Inc., P.O. Pox 2XXA. Ut. 1, Officii, Utah
Rocket Development Corp., Box 522, Rich mood, bid.
Cnited Scientific Co., Inc.. P.O. Box S9, Waupaca, Wis.
FILM
The Itislor,/ ami Development of the fiwkri. 10 min (1962). MeUruwllill
TextFilm Division, 330 West 42 St., New York 36, N.Y. Available
in color or black and white.
38 Looking Out: Rockets, Satellites, Space Travel
SUGGESTIONS FOR FURTHER READING
Hobbs. Marvin: "Fundamentals of Rockets, Missiles, and Spacecraft,"
John F. Rider, Publisher, Inc. New York, L962, 27ft pp.
"An Introduction to Rocket Missile Propulsion," Rocketdync, Canoga
Park, Calif.. 1958, 12ft pp,
"Model Kits," Revcll, Inc., 4223 Cileneoe, Venice, Calif.
"The Next Ten Years in Space, 19591989," House Document 115,
U.S. Government Printing Office, Washington 2ft, DC., 1959,
221 pp. ,
"1959 Missiles and Rockets Encyclopedia," Re veil, Inc., Venice,
Calif., 32 pp.
"Physical Data, Constants and Conversion Factors," General Electric
Missile and Space Vehicle Department (Mail: P.O. Box Sftfto,
Philadelphia 1, Pa.), 1959, 24 pp.
•■Rocket Experiment Safety: Safety Suggestions for the Rocket Hobby
ist/' Atlantic Research Corp., Alexandria, Va., 1958, 19 pp.
Seiferl, Howard S.. Mark M. Mills, and Martin Summerlield: The
Physics of Rockets, American Journal of Physics, 15:121, 121140,
255272 (1947).
"Space Facts: A Handbook of Basic and Advanced Space Might and
Environmental Data for Scientists and Engineers," General Elec
tric Mis>ile and Space Vehicle Department, Valley Forge Space
Technology Center (Mail: P.O. Box 8555, Philadelphia 1, Pa.),
I9f>0. fil pp.
QUESTIONS AND PROBLEMS
1. Verify the statement: "Near the surface of the earth, gravity
robs a vertically rising rocket of about 20 mi/hr in speed each second,
or about 2,400 mi/hr for each 2 iniu of acceleration." ^
2. A projectile is Srcd with a speed of 300 ft/sec at an angle ot 87
with the horizontal. Compute the speed when it first reaches a height of
.jqq fj._ j4«s. 83.5 ft, sec
3. A force of 4,900 dynes acts on a 20gm mass for 8.0 sec. (a) What
acceleration is caused? (I>) Bow Tar docs the mass move from rest in the
8.0 sec? («) How fast is it going at the end of S.O see?
Am. 24ft em/see, 8,140 cm, 1,900 era/see
4. Show that the mass ratio tin/m,, for a multistage rocket is the
product of the mass ratios of its individual stages.
5. Would you consider an alkali metal, such as cesium, a prospec
tive propellant for an ion rocket? Why?
6. If it becomes possible to convert nuclearfusion energy of a plasma
directiv into electric energy, without the conventional rotating gener
ator, would this make ion propulsion of rockets more feasible?
Rocket Propulsion 39
7. Show that if one increases the exhaust temperature of a hcal
cNclianger nuclear rocket, the specific impulse and power requirements
will increase as J' ! and the mass ratio will decrease as exp T. What
limits this favorable picture?
8. The Atlas 1CBM is called a I 'stage rocket because of its
unique application of the step principle. The Atlas has three main
engines: two booster engines and one suslainer. Each engine receives
propellant from a single very lightweight tank. The three engines are
mounted parallel to one another. All three engines are ignited at take
ofT. I/titer. at staging, the boosters ami housing slide backward on rails
and drop to earth, leaving the suslainer engine to propel (he vehicle.
Can you suggest some advantages of this type of staging over the con
vent iona! tandem staging?
9. How can rocket action be demonstrated with a toy balloon?
10. What is the fallacy in the following argument? "A horse pulls
on u cart. By Xewton's third law, the cart pulls back on the horse with
a force etjiuil to that exerted by the horse on the earl. Hence the sum of
the forces is zero, from which it follows that it is not possible for the
horse to accelerate the cart."
11. Comment on the remark, "Space stations will be obsolete when
they are feasible."
12. A rocket whose thrust is 27,000 lb weighs initially 22,000 lb, of
which SO per cent is fuel. Assuming constant thrust, find the initial
acceleration and the acceleration just before burnout. Xeglect air
resistance and variation of g. Arts. 39.2 ft/sec 2 , 190 ft/sec 2
13. From Eq. (3.19) show that for a rocket launched horizontally
and continuing in a path parallel to the earth, the burnout velocity is
given by Vt = err In R.
II. What is the minimum value of mass ratio R for which the
burnout velocity n, of a rocket will exceed the effective exhaust ve
locity iv„? 4 ns. R > 2.718
15. Do you agree with Professor Fink's comment that Hie methods
of achieving lift listed in order of increasing sophistication of the under
lying physical principle are (1) satellite vehicles, (2) displacement of
lighterthanair craft, (3) hover craft or groundcushion vehicles,
(4) vertical flight rockets, (5) vertical takeoff and landing machines,
(<>) conventional airplanes? If so, how do you account for the historical
fact that the "simplest" methods were not the first to be suecessfullv
Used?
Escape
from Earth
No thing is too high for the daring of
mortals : We storm heaven itself in our
folly. Horace
Every great advance in science has
issued from a new audacity of imagi
nation. John Dewey
Through the ages men have dreamed of the power of flight. In
Creek mythology Daedalus and Icarus made a daring ascent
into the air on wings made of birds' feathers and wax. In the
notebooks of Leonardo da Vinci are found detailed drawings of a
flying machine. With the success of the Wright brothers, man
began to realize his long ambition of flight through the air. But
now he turns his dreams to flight beyond the enveloping and
protective atmosphere — into space.
In designing a vehicle to escape the earth, one has to solve the
problem of piercing the earth's atmosphere. A second, more
troublesome, problem of escape is that of overcoming the force
of gravity. Since each body in the universe has its own gravita
40
Escape from Earth 41
tioual field, a vehicle in space would encounter an endless mixture
of gravitational fields, one superposed on another. The terms
escape and capture refer to the transfer of the vehicle from one
field to another.
4.1 GRAVITY
In addition to the three laws of motion, Newton formulated the
law of universal gravitation: Every particle in the universe
attracts every other particle with a force that is directly propor
tional to the product of the masses of the two particles and in
versely proportional to the square of the distance between their
centers of mass
F = G
(4.1)
where F = force of attraction
m [ and mi = masses of the two particles
s = distance between them
G = gravitational constant, whose value depends on
the system of units used
In mks units
G = (5.07 X 10 " newtonmVkg s
Gravity acts as a brake on a vehicle leaving the earth. While
traveling in space, a vehicle is always subject to some gravity.
The vehicle attracts and is attracted by all celestial bodies. But
because gravitational force follows an inverse square law (/•' <*
1 «), the mutual attractions of only the nearest bodies are usually
significant. When a vehicle returns to earth, it is accelerated by
an increasing gravitational force.
4.2 FREELY FALLING BODIES
An unsupported body starting from rest near the surface of the
earth drops 10 ft during the first second, 04 ft at the end of the
iH'Xt second, 144 at the end of the third, etc. It has an acceleration
r, f "V2 ft per sec per sec, or 32 ft/sec. The symbol g is used to
represent the acceleration due to gravity. At sea level and 4:">°
latitude, g has a value of 32.17 ft/sec 2 or 9.806 m/sec 2 .
The value of g varies slightly over the earth owing to local
42 Looking Out: Rockets, Satellites, Space Travel
variations in mass distributions and to the fact that the earth
bulges slightly ai the equator. Surface gravity values vwy from
planet to planet owing to differences in mass, radius, and rota
tional speed.
When air resistance can be neglected, the equations in Sec. 3.2
for uniformly accelerated motion apply to falling bodies.
t, sec
1
v, ft/sec s, ft
O
O
°\
32
64
96
128
16
64
144
256
Fig, 4.1 Position and speed of a body
falling freely from rest after successive equal
time intervals.
4.3 GRAVITATIONAL FIELDS
The force which one body exerts on another at a distance is con
veniently described by the "force field" set up by one of the
bodies, the source. Various kinds of forces can be treated in this
way. Electric charges exert forces upon other electric charges.
Magnets exert forces on other magnets. Matter exerts gravita
tional force upon other matter.
The force exerted on a unit test particle (unit charge, unit
mass, etc.) has a definite magnitude and direction for each pos
sible location of the test particle. The whole assemblage of these
for08 vectors, or the mathematical function relating force to
position, is called a field of force. Any path that would be fol
lowed by a free incrtialess test particle is called a line of force.
The gravitational field intensity / at any point A in the
space near a mass ,1/ is defined as the force per unit mass acting
on any mass m placed at A :
/ =
m
(4.2)
Escape from Earth 43
The small mass m is used only as a means of detecting and measur
ing the gravitation field, Whether m is large or small, the. force
per unit mass placed at .4 has a definite value, J.
By substituting Eq. (1.1) for F in Kq. (4.2), we get the
expression for the gravitational field intensity / at a distance r
from niitss .1/ in terms of the universal gravitational constant Gas
m r 2
(4.3)
When one knows the field intensity, one can find the force acting
on any mass m as the product of m and /,
F
Fig, 4.2 Gravitational forces of attraction.
Fig. 4.3 Parabolic poth of a projectile in a uniform gravitational
field.
For a freely falling body, Newton's second law of motion
becomes F = mg. At the earth's surface, therefore, the gravita
tional field intensity is equal to g, the acceleration due to gravity
' =  = ?
m
I newton
(4.4)
meter
kilogram second 2
111 mka units, g = 9.80 m/sec 2
44 Looking Out: Rockets, Satellites, Space Travel
In a region of free space where the gravitational field is prac
tically constant in direction and magnitude, the path taken by a
projected mass m is a parabola (Kig. 4.3).
4.4 GRAVITATIONAL POTENTIAL ENERGY
To find the work needed to get off the earth, let us calculate the
work done in moving a mass m from the surface of the earth,
radius R, to a distance r from the center of the earth. Imagine
Distance
r * measured
_ from C
Fig. AA Calculation of gravitational poten
tial energy.
the distance from 5 tor to be divided into small equal intervals so
that over each the gravitational force F a will lie practically con
stant. Then we can easily calculate thr work done in each interval
and add to get the total. At the surface, Fa = QMjm A' 1 '. At the
top of the first interval Fa is OMjm r,«. Since these values are
nearly the same, we can use for the average force in the first
interval (!M<w ltr u The work done in the first interval is then
Wt = Fair, ~ 8) = Vg fir.  *> = W* (l " ,')
Likewise the work in the second interval is
„,. 9£ h „«.„.„, (iI)
Escape from Earth 45
and in the third
Wt = GMjn (  )
If we add these three expressions, the intermediate values r, and
n cancel out. The work done in the first three intervals can be
expressed in terms of the values of r at the ends: It and r 3 . Thus
W = GM
 m (it ~ ' )
(4.5)
is the general expression for the work required to move a mass in
against the earth's gravitational field out to a distance r. By
definition this is the gravitational potential energy of mass m in
the field of the earth.
4.5 VELOCITY OF ESCAPE, FROM CONSIDERATION
OF ENERGY
To estimate the maximum height attained by a rocket fired
straight up, we may equate its kinetic energy at burnout to the
gravitational potential energy it acquires thereafter in rising,
with decreasing speed, to its maximum height, t x
^mv 2 = mgR*
\lt ''i..:. </
(4.0)
Example. A roirkci has an upward speed uf 5.0 mi/see :it burnout. Find
the maximum height it attains.
From Eq. (4.6)
mfn mi Y m ft (*ixm ■ 3960 mi' \
V 880/ "»* \ r.„n< )
25 / miV / 32 \ .. i(irin { iniV ( 82 \ /39(i0= mi A
%\&) =Uoj c ^ o) tcj U»/U.*W
r n , ax = 8,S(W mi
(What amplifying assumptions have been made in this Bohition?)
4 6 VELOCITY OF ESCAPE, FROM CONSIDERATION OF FORCE
If a gun on a cliff overlooking the ocean fires a bullet horizontally,
the bullet will strike the water at some distance from the base of
■he cliff. If the initial speed of the bullet is increased, the range
46 Looking Out : Rockets, Satellites, Space Travel
is increased. For a particular speed, which depends on the dis
tance of the gun from the center of the earth, the hullet would
make a complete circuit of the earth, at a constant altitude, A
(Fig. 4,">), If it did not encounter resistance, it would continue
to move in orbit about the earth.
Prom Newton's second law, F = ma, and Eq. (4.2), the
force needed to hold the bullet in a circular path at altitude A is
Fig. 4.5 Range increase* with horizontal firing speed until circular (orbital
speed is reached.
invfQi + R). This force is provided by the gravitational attrac
tion of the earth, so
/' 
CM ,i» ini
(/,' + /,)■' H + h
(4.7)
Example. At what speed would a projectile have to leave a platform,
horizontally, 300 mi above the earth in order to enter a state of "con
tinuous fall" around the earth?
From Kq. (4.7),
C.U e 6.67 X IP"" newtonm* 5.983 X 10" kg
ys = 
R + h kg'
= 58.1 X ICm'/sec*
w = 7,«2() in sec = 1,700 mi/hr
mi
(3,950 + 300) mi 1,609 m
4.7 WEIGHTLESSNESS
A body in orbit around the earth or following an unrestricted,
un powered course in a gravitational field anywhere in space is
said to be in "free fall," also called a state of "zero gravity."
Actually, gravity is not absent. The force of gravity continually
Escape from Earth 47
acts on the body and determines its path. But the condition of
weightlessness is experienced because there is nothing to resist
the body's motion in response to gravity.
Human beings have experienced weightlessness for the first
few seconds after leaving a high diving board, or for somewhat
longer periods in aircraft on "zero g" trajectories, and more
recently in manned rocket flights.
4.8 POTENTIALWELL MODEL
Using Eq. (4."»), we may plot a graph showing the potential
energy K p which a body of mass ni would have at various dis
tances r from the center of the earth (Kig. 4.6). When the mass m
50,000
25.000
,,„, ennnn ■ S GrOVltOtlOnol
25,000 50, 000 mi f ,
1 1 £■ free •■
: spoc
Fig. 4.6 Gravitational potential energy of mass m, showing "well" analogy for
earth's field,
is infinitely far from earth, li p = 0. As mass m is brought closer
and closer to the earth, work is done on m by the earth's field and
the potential energy of m acquires a larger and larger negative
value. Thus on the surface of the earth we live in a gravitational
well thousands of miles deep. To reach the moon or another
planet we must climb out of this well onto the plane marked
"gravitational free space" in Fig, 4.(i.
A potcntialenergywell model for demonstrating satellite
orbits may be made from a suitably shaped wine glass to reprc
48 Looking Out: Rockets, Satellites, Space Travel
sent the surface (Fig. 4.7) obtained by rotating the graph of Fig.
4.6 ahout its vertical axis. A marble representing the satellite
may be caused to travel a variety of orbits by varying its initial
velocity.
Fig. 4,7 Potential well model for demonstrating circular
(c) and elliptic (e) orbits. ({See J. S. Schooley, Satellite
Orbit Simulator, American Journal of Physks, 30: 531532
(1962).]
QUESTIONS AND PROBLEMS
1. What is the largest gravitational force of attraction between
two solid metal spheres cadi of 50.0 kg mass and 10.0 em radius? How
does lllis force compare with the force of attraction of the earth on
each sphere?
Ans. 4.17 X 10"° newton, weight is 120 million times larger
2. What would be the value of <j. the acceleration clue to gravity,
if the earth had half its present diameter?
3. If the mass of the moon were doubled hut the orbit remained
the same, what would he the period of the moon?
t. A 100lb man starts sliding down a rope with a downward
acceleration of p/S. (a) What is his apparent weight? (b) What is the
tension in the rope above the man?
5. Using the experimentally determined value of (7 and the distance
93 X 10° mi from carlh to sun. calculate the mas* of the sun.
6. At what point in its trajectory does a projectile have its mini
mum speed?
7. If a rocket at tains a speed of (500 mi hr hy the time it reaches
1,000 ft, how many times g is its acceleration?
8. The earth revolves about the sun in a nearlv circular orbit
Escape from Earth 49
(r = 150 X 10 e km) with a speed of about 30 km see. What is the
acceleration of the earth toward the sun?
9. Show that to escape from the atmosphere of a planet, a molecule
of gas must have a speed r such that t ! > 2C.i//r, where .1/ is the mass of
the planet and r is the distance of the molecule from the center of the
planet. What hearing does this have on the composition of (lie atmos
phere surrounding the earth and other planets?
10. A balloon which is ascending at the rate 12 m/sec is 80 m above
the ground when a lour is dropped from it. How long a time will be
required for the stone to reach the ground? Ans. 5.4 sec
11. An elevator is ascending with an acceleration of 4.0 It sec 2 . At
the instant its upward speed is K.O Ft/see, a holt drops from the top of
the cage 9.0 ft from its floor. Find the time until the holt strikes the
floor and the distance it has fallen. Ans. 0.71 sec, 2.3 ft
12. A body hangs from a spring balance supported from the roof
of an elevator, (a) If the elevator has an upward acceleration of 4.0 ft/
sir and the balance reads 45 b, what is the true weight of the body?
(ft) In what circumstances will the balance read 35 lb? {<■} What will the
balance read if the elevator cable breaks?
Ans. 40 lb, a = 4.0 ft/sec s downward, zero
13. If the mass of the moon is ^ the mass of the earth and its
diameter is J that of the earth, what is the acceleration due to gravity
on the moon? How far will a 2.0 gm mass fall in 1.0 .cc on the moon?
Ans. I g, 3.2 ft
14. A girl standing on a diving hoard throws a ball with a hori
zontal velocity of 50 ft/sec to a man in the water. In doing so, she loses
her balance, falls off the hoard, and strikes the water in 2.0 sec, (a) How
far is the man from the base of the diving board? (b) How high is the
diving hoard above the water? (e) What is the velocity of the ball at
the end of its path?
.bis. 100 ft, 64 ft, SI ft/scc, at 52° with the horizontal
Satellites 51
Satellites
It is no paradox to say that in our most
theoretical moods we may be nearest
to our most practical applications,
A. N. Whitehead
The launching of the first artificial earth satellites, the Russian
Sputniks I and II, in 1957, aroused worldwide interest in the
power and control attainable with rockets. The special scientific
investigations made during the International Geophysical Year
were significantly aided by data from instruments carried in
satellites. From the orbit of a satellite one may better estimate the
shape and dimensions of the earth. A permanent satellite can be
useful as an aid in the navigation of ships, aircraft, and missiles.
For a satellite which eventually returns to earth, measurements
of the orbit may yield a more precise value of g. Atmospheric
drag and the effectiveness of radio emission at various altitudes
can be studied. Equipped with suitable instruments, a satellite
can also measure solar and cosmic radiation, temperature and
pressure variations, and the distribution of the earth's magnetic
field. In short, satellites can tell us much that we want to know
about our earth and much that we need know about space hazards
before we venture into space ourselves.
50
5.1 ELEMENTS OF AN ORBIT
To define the position of an earth satellite in the solar system and
to describe its path, one needs to know the period of the satellite
and the elements of its orbit, that is, the constants which fix its
position and shape in space:
The period is the time for a satellite to make one revolution
around the earth.
The perigee is the position of closest approach to the center
of the earth. Apogee is the position of the satellite farthest from
the earth (Fig. 5.1a),
The eccentricity describes the flatness of the orbit as the ratio
of e to a (Fig. 5.1/j). Here e is the distance from the center of the
Apogee
Perigee
Fig. 5.1 Elliptical orbit.
orbit to the focus at the center of the earth, while a is the semi
major axis.
The angle of inclination i of the orbit is the angle between the
plane of the orbit and the plane through the equator (Fig. 5.2).
The plane of the satellite orbit intersects the equator plane in
a straight line called the line of nodes. This line intersects the
satellite orbit at two points, called nodes. At one of these, the
ascending node, the satellite crosses northward from "below"
the equator plane to "above" the equator plane. At the other,
the descending node, the satellite crosses southward from "above"
th f equator plane to "below" the equator plane. The orbit ele
ment we now define is the longitude V, of the node, or the angle of
"■•<>•' tiding node. This angle, P. in Fig. ,5.2, is measured in the plane
J f the equator from the direction of the vernal equinox to the
unction of the ascending node. (To describe the motion of a
Q
o
Satellites 53
planet about the sun, one substitutes "ecliptic" for "equator"
in the definitions above.)
The argument of perigee us is the angle measured in the orbit
plane between the direction of the ascending node and the direc
tion of the perigee.
To summarize, the elements of an orbit are period, perigee
apogee, eccentricity, angle of inclination, angle of ascending node,
ami argument of perigee.
5.2 CIRCULAR ORBIT
The path of any body acted on only by an inverse square force
(/' a l/r ! ) due to a neighboring fixed body will ho an ellipse,
circle, parabola, or hyperliola (Chap. (i). To simplify our analysis,
for the remainder of this chapter we shall examine an earth
satellite in a circular orbit and consider only the interaction be
tween the earth and the satellite. Although small perturbations
may be produced by the atmosphere, the moon, other planets,
and satellites, for the present these effects will be neglected.
For a satellite in circular orbit, the gravitational force exerted
on it by the earth has no component in the direction of motion
which could either increase or decrease the speed of the satellite.
It orbits at constant speed.
The force on the satellite is given by Xewton's law of
gravitation
F = G
Mm
(5.1)
where the mass of the earth M and the mass of the satellite m are
icffiirded as concentrated at the center of each, a distance r apart.
The constant of gravitation G can In* determined in the labora
tory. Because the mass of the earth is so very large, the center of
mass of the two bodies is practically at the center of mass of the
earth. The motion may be described as a circular motion of the
satellite about a fixed center of force.
The direction of the velocity of the satellite in circular orbit
» continually changing (Fig. 5.3). Gravitational force continu
ally produces a "centripetal" acceleration a toward the center.
54 Looking Out: Rockets, Satellites, Space Travel
Satellites 55
For this uniform circular motion the acceleration is v/r and the
centripetal force is mv*/r.
/ y — \A
v ¥ canst.
\13\ const.
\a\— const.
Fig, 5.3 A satellite in circular orbit is continually
accelerated toward center of orbit.
5.3 PERIOD
We may equate the gravitational force and the centripetal force
f ,Mm _ mv' 1
G — z — ~rr
(5.2)
From this equality we find the speed that a satellite has to obtain
to maintain a particular altitude
GM
v = x 
(5.3)
The mass m of the satellite does not appear in the equation for
speed. The closer the satellite is to the earth, the greater must be
the speed, because the gravitational attraction is greater.
Since the angular speed tn is 2w/ "period,
2x JOM
and since
we obtain an equation.
7'2 =
GM
(5.4)
which says that the square of the period of the satellite is propor
tional to the cube of its distance from the center of the earth.
This is Kepler's third law of motion, for the special case of a
circular orbit.
5.4 ENERGY
The total energy remains constant in satellite motion. This can
be shown very easily for our special case of a circular orbit.
Substitution for the speed of the satellite (Eq. 5.3) into the equa
tion for kinetic energy E k gives
v i— * GMm
h k = \mv* =  2r
(5.5)
Since the gravitational potential energy Ey of the satellite is
GMm
B, = 
1 1 ie total energy is
E k + E„ «■ 
GMm
2r
(5.6)
(5.7)
The total energy is negative for both circular and elliptical orbits.
This means that the satellite is bound to the center of force and
cannot escape unless sufficient positive energy is provided (see
*'ig. 4.7).
5.5 ANGULAR MOMENTUM
Iho total angular momentum L of a satellite moving at constant
speed in a circular orbit is the product of its linear momentum mv
56 Looking Out; Rockets, Satellites, Space Travel
and the radius r. Vcctorially, the angular momentum is repre
sented by a vector L drawn to scale to represent the scalar mag
nitude mvr and drawn along a line perpendicular to the plane of
r and v in the direction indicated by tlie thumb of the right hand
when the fingers are allowed to curt from the direction of r into
the direction of v. Thus L results from a vector "cross product,"
the notation for winch is
L = rX (hit)
It is obvious that for a satellite in uniform circular motion
the total angular momentum is constant. This is also true when a
*S^
Fig. 5.4 Angular momentum of mass m is re
presented by vector I.
satellite moves in an elliptical orbit. The radius and speed vary,
but the total angular momentum remains constant. This is
equivalent to Kepler's second law, that a line joining the focus
and the satellite sweeps out equal areas in equal periods of time
(Fig. 5.5).
Fig. 5.5 The satellite sweeps out equal
areas in inequal periods of time.
It is not feasible or even particularly desirable to launch a
satellite into a perfectly circular orbit. If such an orbit were
attained, slight perturbations would soon make it elliptical.
Observation of a satellite in orbit gives us information about
irregularities in the shape of the earth. As the satellite orbits, the
plane of its orbit rotates or regresses toward the west. At the
same time the orbit turns in its own plane, swinging the perigee
around. Also, both ends of the orbit become somewhat flattened.
Satellites
57
These observations are interpreted as proof that the earth
bulges slightly around the equator, owing to the earth's rotation.
The gravitational force tends to pull the satellite toward the
equator. Consider the gyroscopic property of the satellite. The
gravitational force due to the bulge tends to tip the axis of the
Earth's rotation
Fig. 5.6 Rotation of the earth and precession of the satellite orbit expose
different areas of the earth to the satellite, os shown in Fig. 5,7.
orbit. The reaction causes the plane of the orbit to prccesK around
the earth in a westerly direct ion, while the earth is rotating from
"est to east. This precession may be an advantage in the case of
Certain types of observational satellites which thus may "see"
most of the earth's surface (Kigs. 5.6 and 5.7).
Satellites 59
QUESTIONS
I. Why is tin* upper (dotted) path in i)k> accompanying sketch not.
a possible satellite orbit about the earth?
Fig. 5.8
2. Show ilial if frictional forms cause ;i satellite to lose total energy,
it will move into an orbit closer to (he earth with an actual increase
in speed.
3. After a certain satellite was put in orbit, it was stated thai the
satellite would not return to earth but would burn up on its descent.
I low is this possible, since it did not burn up on ascent?
Motion of Bodies in Space 61
Motion of Bodies
in Space
If I have seen farther than ethers, it is hy
standing on the shoulders of giants. An
old saying quoted by Newton
A space vehicle when not under power is governed by the same
laws which determine the motions of stars, planets, and comets.
These laws are Newton's law of universal gravitation and Kep
ler's taws of planetary motion. Karly in the seventeenth century,
Kepler by inductive reasoning formulated his three laws to fit
the astronomical observations and calculations made available
1 1! him by his patron Tycho Brahc. Xewton in his "Principia
Mathematica" (Ih'87) showed that the kind of planetary motion
described by Kepler's laws can be deduced from the universal
law of gravitation.
6.1 KEPLER'S LAWS
Kepler's description of planetary motion may be stated as follows;
Law I. The planets move in ellipses having a common focus
situated at the sun.
Law II. The line joining the sun and a planet sweeps out equal
areas in equal periods of time.
Law III. The square of the period of a planet is proportional
to the cube of its mean distance from the sun.
An ellipse may be constructed by using two pins and a loop
of string to guide a pencil (Kig, 0. 1). This method of construction
Fig. 6.1
ellipse.
Construction of an
makes use of a geometrical property of the ellipse: The sum of
the distances from any point on an ellipse to the two foci, A and
li. is constant. An ellipse with its center at the origin of coordi
nates and with foci on the x axis is represented by an equation
<>!' the form
+ £ = 1
a 2 T b*
From Kepler's second law, if the shaded areas in Kig. (>,2 arc
S3
x, f~~^
Fig, 6.2 Law of areas.
a " equal, a planet takes equal time intervals to travel the dis
tances St, $ 2l and s a .
Kepler's third law, called the harmonic law, expresses the
Proportionality of period squared, 7", and the cube of the scmi
Boajor axis a of the ellipse.
60
62 Looking Out: Rockets, Satellites, Space Travel
Example. Calculate the height of a satellite in a 24hr orbit about the
earth if it has been observed that a satellite at a mean distance of
4,100 mi from the center of the earth has a period of 5,000 sec.
From Kepler's third law
3Y
7V
a.*
we wish to find o 2 when T~ = 1 day = 8.6 X 10 4 see
= (I/V a , = [tf)l 4,100 mi = 27,000 mi
ai
Kepler's lawn apply to the ideal ease of only two bodies mov
ing under their mutual gravitational attraction. But in space
travel, effects of other bodies have to he considered.
To consider the feasibility of certain proposals or devices,
one starts by examining qualitative orbits. Such trajectories are
predicted with the aid of simplifying assumptions: that the moon
moves in a circle around the earth, that the earth may be con
sidered symmetrical, that any disturbing masses are in the orbit
plane of the space vehicle, etc. The precision trajectories needed
for actual space travel do not allow these approximations. Hence
the calculations become enormously more complicated.*
6.2 NEWTON'S DERIVATION OF KEPLER'S LAWS
As a test of his theory of universal gravitation, Newton desired
to show that Kepler's laws could be derived from the law of gravi
tation and he desired to investigate the more general problem:
What kind of motion is necessary according to that law? In its
basic statement, the law of universal gravitation applies only to
particles ("point" masses). Newton needed first to show that the
attraction for an exterior particle exerted by a spherical mass
(either homogeneous or somewhat like the earth, made up of
concentric homogeneous shells) was directly proportional to the
total mass of the sphere and inversely proportional to the square
of the distance of the particle from the sphere's center. Newton's
difficulty in establishing this principle to his satisfaction may have
been the cause of his delaying some twenty years in publishing
his conclusions.
* Precision rocket orbits are discussed in S. Herrick, "Astrody
namies," I). Van Xostrand Company, Inc., Princeton, N'..l., 1959.
Motion of Bodies in Space 63
The orbits of all the planets (except Pluto) are very nearly
circles, with the sun at the common center. Kepler's third law
can be derived by equating the centripetal force to the gravita
tional force (Sec. u.3) to obtain
T* 4» !
i* = m = con8tant
Kepler's second law, the law of equal areas, follows whenever
the interaction between two particles is in the direction of the
line joining them. The force need not follow an inverse square.
Fig. 6.3 Derivation of Kepler's second law.
Let Pi, P>, and P% be points along a planet's orbit marking the
position of the planet at time intervals of 1 sec. Then the distance
PiPz is numerically equal to the planet's velocity 1% and /V J 3 is
numerically the velocity r., in the next second. When the only
force acting on the planet is in the direction of the sun, this force
has a component zero perpendicular to line / J 2 ,S'. Hence the com
ponent of the planet's velocity perpendicular to / J ,.S' must be
unchanged, according to Newton's first law of motion: r tL = u 2l .
The area swept during the first second by the line joining the
planet and the sun is &P1P1. The area swept in the next second
is SP t P 3 . These triangles have the same base P°S and equal alti
tudes v ± ; hence they have equal areas.
The derivation of Kepler's first law is more lengthy, and it
involves differential equations.* The question is: Given an in
versesquare law of attraction, what shape must a planet's (or
comet's) orbit have? The answer turns out to be: The orbit will
be one of the conic sections with the attracting body (sun) in
one focus,
* A derivation without calculus is presented in Jay Orear, "Funda
mental Physics," pp. 7073, .John Wiley & Sons, Inc., New York, 19GI.
64 Looking Out: Rockets. Satellites, Space Travel
Conic sections are curves? obtained by taking plane slicas of a
solid circular cone (Fig. G.4). The cone sliced parallel to its base
(I) gives a circle. If the cut is slanted, the section is an ellipse (2).
With greater slant, the section is a parabola (3). With still greater
slant, the section is a hyperbola (4).
1. Circle
2. Ellipse
3. Parabola (parallel to lineaO)
4. Hyperbola
Fig. 6.4 Basic orbits related to conic sections.
Fig. 6.5 Newton's proposal for an earth satellite, [(a) From Sir Isaac Newton,
"Mafnemoficai Principles . . . ," edited by F. Co/oW, University of California
Press, Berkeley, Calif., 1934. (b) From E. M. Rogers, "Physics for the inquiring
Mind," Princeton University Press, Princeton, N./., 1960.1
Motion of Bodies in Space 65
The significance of the various conicsection trajectories may
be clarified by an example based on Newton's own suggestion for
an earth satellite. About 1660 he predicted in a drawing (Fig,
(5.o) that if a cannon ball could be fired with a muzzle velocity of "i
mi/sec, it would circle the earth as shown. The Sputnik and
Explorer satellites did achieve this velocity, For a lowflying
earth satellite in a nearly circular orbit, equating the centripetal
acceleration r/r to g gives
v = \/gr — 9.8 X (>.' r > X 10 6 m/sec = 8 km/sec or 5 mi /sec
Now alter Newton's drawing (Fig. (U>) by considering the
mass of the earth to be concentrated at point E (Fig. (i.(i). Con
Hyperbola
Fig. 6,6 Conic orbits cotangent at satellite launching point p.
^ider that a satellite is to be launched at point p with a velocity
Perpendicular to the line K v . Jet the, circle represent the orbit of
the satellite described in the preceding paragraph. The effect of the
earth's attraction is to cause the satellite to fall r = Igl 1 = 4.9 m
toward the earth in the same second it travels 8 km along the
tangent. The two displacements bring the satellite back to the
66 Looking Out: Rockets, Satellites, Space Travel
same distance it had before. So, during each second, the satellite
falls toward the earth but never gets any closer.
Now suppose that the satellite's velocity is made less than
S km/sec. The earth's effect of 4.0 m each second is unchanged.
So the satellite will fall closer to the earth along the smaller
ellipse of Pig. 6.0. Since the earth is not a "point" as implied in
Fig. 0.0, the satellite actually will not he able to complete the
elliptical orbit but rather will strike the earth after traveling a
trajectory which is si portion of an ellipse (Fig. 6.5). (The smaller
ellipse of Fig. 0.0 could represent the path of a comet or planet
about the sun at A'.)
If the satellite at p were given a velocity somewhat greater
than 8 km/sec, the 4.9 m by which it would fall to the earth each
second would be insufficient to hold the distance constant. The
satellite would climb away from the earth on the larger of the two
ellipses (Fig. 0.0). With decreasing speed the satellite would
arrive at a point a opposite the start. There the centrifugal
reaction would be insufficient to overcome gravitational attrac
tion, though the latter would also have decreased. Accordingly,
the satellite would begin to fall back toward the earth, regaining
speed along the elliptic path until it reached point j> with the same
velocity as at the start.
Increasing the satellite's velocity at p still more would semi
it off along the parabola shown in Fig. 6.6. Still greater velocities
would carry the satellite away from the earth along a hyperbolic
path. In either case the attraction of the earth would be insuffi
cient to decrease the radial velocity of the satellite enough to
cause it to return.
QUESTIONS AND PROBLEMS
1. The periods of revolution of the planets Mercury, Venus, Mars,
and Jupiter are, respectively. 0.241, 0.017, 1.88, and 11.9 years. Find
Iheir mean distances from the sins, expressed in astronomical units
(1 a.u. = distance from sun 1o earth).
2. .Jupiter lias a radius of 74,000 km. A satellite completes an orbit,
about Jupiter every J 6.7 days. The radius of the orbil of the satellite is
27 times the radius of the planet. Compute the mass of Jupiter.
3. What docs Kepler's second law say about the duration of winter in
the Southern Hemisphere (which occurs in .inly when the earth is farthest
from the sun) compared with winter in the Northern Hemisphere?
4. Show the correctness of Kepler's third law of planetary motion
Motion of Bodies in Space 67
by equating the centripetal force required to keep a planet in its (circu
lar) orbit to the gravitational force due to the sun's attraction.
5. What is the mass of a planet, .1/,., if it is observed to have a moon
revolving about it at a distance /?, center to center, in period T?
6. If the earth, considered to be spherical, were to shrink to 0.9 of
its present radius, what changes would occur (a) in the length of the
solar day, (/>) in the value of g at the North Pole, (<•) in the value of g
at the equator?
7. The earth satellite Kxplorer III had a highly eccentric orbit witli
perigee at a height of 109 mi. At this point the velocity was 27,600 ft/sec
in a direction perpendicular to the radius to the center of the earth.
Show that this speed is too great for a circular orbit at the radius
(R, + h) of 4,109 mi. Hence the satellite described an elliptical orbit.
Its apogee was at the height 1,630 mi. Show that the speed at apogee
was too small for a circular orbit at radius 5,630 mi.
It is the supreme art of the teacher to awaken joy in creative expression and
knowledge.
A. Einstein, motto for the Astronomy Building, Pasadena Junior College
Putting on the spectacles of science in expectation of finding the answer
to everything looked at signifies inner blindness. J, Frank Dobie
Science is built up with facts, as a house is with stones. But a collection of
facts is no more a science than a heap of stones is a house. H. Poincare
Science is organized knowledge. Herbert Spencer
Science is nothing but perception. Plato
Travel to Moon
and Planets
We first throw a little something into
the skies, then a little more, then a
shipload of instruments then our
selves. . Fritz Zwicky
The solar system, consisting of 9 planets moving in elliptical
paths around the sun, 31 known moons, and many other bodies
all in motion, does not invite simple straightline travel from the
earth to a selected destination. To conserve both power and time,
departure dates and trajectories must be chosen which utilize
favorable positions and relative velocities. Conditions favorable
for return passage may not occur until some time later.
Owing to the everchanging distribution of bodies in the solar
system, no two courses between even the same two bodies are
likely to be the same. The calculations of desired trajectories
and corrections of the course while in flight are complex tasks
for computing machines.
7.1 INVITATION TO INTERPLANETARY FLIGHT
Despite obvious difficulties of travel in the solar system, there are
some interesting favorable factors. The space between the earth
68
Travel to Moon and Planets
69
and other bodies in the solar system is almost a perfect vacuum.
This is an ideal environment for a space vehicle to move at speeds
which make it practical to travel interplanetary distances. Since
I he earth is one of the smaller planets, it requires a comparatively
low escape velocity. Its relatively thin atmosphere offers less
resistance to rapidly ascending and descending objects. The fa el
that the planets lie in nearly the same plane and move in the same
sense makes it possible for an interplanetary traveler to apply the
orbital speed of one planet in launching himself to another. The
fact that the elliptical orbits of the planets are nearly circular
means that the energy requirements to transfer a spaceship from
one orbit to another do not vary greatly for different points of
departure along the orbit. Finally, most planets rotate in the
same direction in which they revolve about the sun. So a space
ship launched at the surface of one of these planets can get an
added push by taking oil" in the direction of rotation.
7.2 LAUNCHING
Before it is launched, the space vehicle is at the earth's distance
from the sun, and it is moving with the earth's speed around the
sun (about 100,000 ft/sec). If launched at greater than the
earth's escape velocity, the vehicle will take up an independent
orbit around the sun, at a speed somewhat different from that of
the earth.
Fig. 7.1 Launching! to inner and to outer planets.
70 Looking Out: Rockets, Satellites, Space Travel
If it is launched in the same direction as the earth's orbital
motion, the vehicle will have a speed greater than that of the
earth (l'"ig. 7,1.1), and could reach the outer planets, Mai's,
Jupiter, etc., if properly directed. The minimum starting speeds
required to reach these planets arc given in Table 7.1.
Table 7.1 Minimum launching speeds, with transit times to
reach the planets
Plonet Minimum launching
speed, ftsee
Transit time
Mercury
44,000
1 10 days
Venus
38,000
150 days
Mars
38,000
260 days
Jupiter
46,000
2.7 years
Saturn
49,000
6 years
Uranus
51,000
1 6 yea rs
Neptune
52,000
31 years
Pluto
53,000
46 years
From R. W. Buchheim, "Space Handbook," 1958.
If the vehicle is launched "backward," against the earth's
velocity, it will move in an orbit like H in Fig. 7.1, so it could reach
VettUS or Mercury. However, it requires almost, as much energy
to propel a vehicle in to Mercury as to propel it out to Jupiter.
7.3 ROCKET GUIDANCE
In the flight of an unmanned probe, satellite, or missile, one or
more boosters provide the initial impulse, but after burnout the
remainder of the flight is unpowered. The vehicle coasts in the
complex gravitational field of interplanetary space. The accuracy
of guidance is generally determined by the position and velocity
at the instant free flight begins. Figure 7.2 gives an idea of the
maximum allowable errors of angular alignment, and vehicle
velocity at power cutoff for several kinds of moondirected
misions. Inertiul Isiyro) or radioguidance techniques are ade
quate for such relatively simple missions.
Interplanetary expeditious present complex problems of guid
ance. First, a launching site might be chosen at not more than
2:}° north or south latitude. This is the angle of inclination of the
ecliptic plane to the earth's equator. The vehicle would be
launched into a satellite orbit around the earth, in the ecliptic
Travel to Moon and Planets 71
S i.o
§
>
'G 1
_o
«
>
o a,
1= I 0.1
 S.
.8 o.oi
E
§
o>
0.001
Scientific sotellire (±100 mi)
Impact on moon (+100
Around moon
return to
braking eclipse
0.001
0.01 0.1
Speed error, per cent
Fig. 7.2 Maximum permissible errors for alignment of velocity vector and for
speed at power cutoff. (Genera' Electric, "Space Facts.")
Initio! ascent ^ ^~v •'/ Earth '
^^
Fig. 7.3 Possible flight paths to Mars,
plane, >'ig. 7.H. With the vehicle in orbit, an ionpropulsion sys
tem might be started to cause the vehicle to spiral out into a
legion where the sun's gravitational field is stronger than the
72 Looking Out: Rackets, Satellites, Space Travel
earth's. The vehicle would then be guided into an elliptical trans
fer orbit around the sun, planned to intercept the orbit of the,
destination planet. Where these orbits intersect, the vehicle
would be directed into an orbit around the destination planet.
Radio or inertial guidance techniques could serve in the early
stages of such an interplanetary flight, but would probably
be inadequate for interplanetary missions of a year or more in
duration, A threedimensional form of presentday twodimen
sional celestial navigation may be necessary. A useful instrument
for establishing a reference direction is the horizon seeker which
senses the infrared radiation of the earth or some other warm
body. Optical trackers and magnetometers may also provide
data to establish the vehicle's position.
To orient and stabilize a space vehicle, torque is produced,
either by the ejection of mass (rocket exhaust) or by the rotation
of a mass within the vehicle. The internal type of torque control
serves to rotate the vehicle about its center of mass ; it does not
influence the flight path.
7.4 RETURN THROUGH THE ATMOSPHERE
To return safely to earth, a space vehicle must overcome the
problems of penetrating the earth's atmosphere. There are three
general types of reentry path, each with its characteristic de
celeration pattern: direct descent, orbit decay, and lifting descent.
These are illustrated in Kig. 7.4. In direct descent into the atmos
phere, the maximum deceleration experienced is independent of
the drag characteristics of the vehicle, but depends on the path
angle, initial velocity, and characteristics of the atmosphere.
The altitude at which maximum deceleration occurs does depend
on the drag characteristics of the vehicle.
For entry of the atmosphere in orbit decay, the vehicle exe
cutes many revolutions about the earth in a very gradual spiral
that becomes more and more nearly circular. The rate of energy
loss through aerodynamic drag is sufficiently small so that the
vehicle's kinetic and potential energies adjust to a momentary
"equilibrium" orbit, with potential energy decreasing and kinetic
energy increasing. Thus the velocity of the vehicle actually in
creases in the start of orbit decay. The final phase of descent is
similar to that of direct descent at a shallow angle.
In a lifting descent, the aerodynamic characteristics of the
Travel to Moon and Planets
73
vehicle are used to obtain a very gradual penetration of the
atmosphere. The path angle is generally small, a few tenths of a
degree, and is adjusted to the forces acting on the vehicle. Decel
eration increases gradually and can be limited to a relatively
small value.
The more gradual the descent, the longer is the time required
and the longer is the range. Starting at a given altitude and
velocity, a direct descent may traverse a distance of a few hun
dred miles and require about A min. An orbit, decay might cover
a range of several thousand miles and require 5 to 10 min. A
Ballistic rocket
Direct from space
Fig. 7.4 Different types of atmospheric entry. (Generaf Elec
tric, "Space Fads")
lifting descent from the same point might range over o,000 to
10,000 mi and require 2 hi*.
When a vehicle penetrates the atmosphere, the reduction of
the vehicle's energy is accompanied by an increase in the thermal
energy of the surrounding air, some of which is communicated to
the surface of the vehicle. At very high altitude, about onehalf
the energy loss appears as heat in the body. At lower altitudes,
the heating is produced not directly at the vehicle's surface but
in the air between the shock wave and the vehicle. Heat is trans
ferred from the hot gases of this region to the vehicle by conduc
tion, convection, and radiation.
7 5 THE NEXT DECADE OF SPACE RESEARCH
Historically, man's attempts to predict the future of science and
technology have shown a tendency to be overly optimistic about
74 Looking Out: Rockets, Satellites, Space Travel
what will bo accomplished in the immediate future and too con
servative about the longrange future.
It has been predicted that man's curiosity about the unknowns
of outer space can be only partly satisfied by the placing of
meters in outer space; eventually he will want to go there to see
for himself. But it is probable that the extent and pace of space
research in the foreseeable future will be determined by what
are regarded as our military requirements.
The military advantage to be gained from putting man in
space is at least debatable, From the standpoint of psychological
warfare, there may be better ways of demonstrating our scientific
prowess. For man's future happiness, more important pure
science experiments might be performed in other fields, such, as
medicine. Yet many dedicated scientists feel that maninspace
experiments are important to our chances of survival
This viewpoint is stated by Dr. Simon Ramo in the following
terms. Suppose two rival nations base their security on a race
for wisdom in the use of limited technical and physical resources.
Suppose, however, that the first nation makes one decision in
contrast to that of the second: It decides that man will never be
needed in space. These two nations start to develop their weapons
systems of the future. One group has maximum flexibility; the
other has some prohibitions. "To achieve this maximum of
flexibility, it is very clear to me that the United States must
prepare for putting man in space," says Dr. Ramo.
On the assumption, then, that we shall have military
sponsored programs in space technology, one can make some
predictions for the near future. Many projects involving com
munications, weather prediction, manned satellite stations, and
exploration within the solar system will probably be fulfilled.
Exploration beyond the solar system now seems unattainable
chiefly from considerations of time and power. Man's life is short
when compared with the time required to reach the nearest star,
even in a vehicle traveling with a speed approaching that of light.
The other problem, "Where is the energy In he obtained for long
voyages or to lift large masses into space?" may find an answer in
the achievement of a nuclearfusion reactor. It is the thermo
nuclear bomb which threatens to make the earth a very unpleas
ant place. Ironically, the energy of a controlled thermonuclear
reaction may provide us with the power resources for a migration
into space.
PART
Looking In:
Atomic and
Nuclear Physics
8
The Atomic Idea
Science does not know its debt to imagi
nation, R. W. Emerson
Although Democritus had introduced the word "atom," it was
the English school teacher John Dal ton (1803) who made fertile
the assumption that matter is not divisible indefinitely but rather
is composed of ultimate particles called atoms. Physics dealing
with phenomena on a scale large enough to be visible to the un
aided eye was well understood by the year 1890. Then a remark
able mutation occurred in science, caused by the series of dis
coveries made in the decade 18951905:
1895 Discovery of x rays by Roentgen
18% Discovery of radioactivity by lleequerel
1897 Identification of the electron by Thomson
H'OO Statement of the basic postulate of quantum theory by
Planck
Pi05 Formulation of the theory of relativity by Einstein
It became clear that the structure of matter was much more
complicated than had previously been thought.
The term "modern physics" is often used to designate micro
scopic (atomic and nuclear) physics, investigated from the view
77
78 Looking In: Atomic and Nuclear Physics
point of quantum theory and relativity, as distinct from the
macroscopic or "classical" physics which was known before 1890.
8.1 DEVELOPMENT OF THE ATOMIC CONCEPT OF MATTER
The existence of atoms has been inferred from many experiments,
the earliest of which were studies of simple chemical professes.
By 1800, some 30 elements had been identified and the formation
of chemical compounds had been studied. Lavoisier showed that
mass appeared to be conserved in chemical reactions.
Proust, Dalton, Berzelius, and Richter discovered "laws"
which may be summarized in the statements:
1. A particular compound always contains the same elements
chemically united in the same proportions by weight. (Law of
definite proportions.)
2. When two elements A and B combine as constituents of more
than one compound, the weights of B which unite with a
fixed weight of A (and vice versa) aje related to each other as
the ratios of whole numbers, which are usually small. (Law of
multiple proportions.)
Dalton showed that these chemical laws could be explained
most directly in terms of an atomic theory of matter. Its assump
tions arer
1. All matter is made up of elementary particles (atoms) which
retain their identity in chemical reactions.
2. The atoms of any pure substance (element) are alike (on the
average, at least) in mass and other physical properties,
:>. Atoms combine, in simple numerical proportions, to form com
pounds.
Dalton's clear formulation of the atomic concept of matter is the
first important landmark in the development of modern atomic
physics.
8.2 AVOGADRO'S NUMBER
Joseph GayLussac (1808) showed that, at a constant tempera
ture and pressure, gases combine in simple ratios by volume.
Amadeo Avogadro was led (181 1) to make the important assump
tion that equal volumes of different gases under the same coudi
The Atomic Idea
79
tions of temperature and pressure contain the same number of
molecules. This hypothesis guided Bcrzelius and others in deter
mining the ratio of combining weights (e.g., is water HO or H s O?).
A molecule is the smallest particle of any substance (element
or compound) as it normally exists.
An atom is the smallest portion of an element found in a mole
cule of any of its compounds. An atom is the smallest portion of
an element that can enter into chemical combination.
By measuring combining weights, it is possible to determine
the relative masses of atoms of various elements. We may arrange
them in order of increasing mass, assigning a number to each to
indicate its relative mass. Since only the ratios of the numbers are
important, we may assign one number arbitrarily to a particular
atom and adjust the others accordingly. Conventionally, the
number Hi (exactly) was assigned to an oxygen atom. Then by
Avogadro's hypothesis, for any gaseous substance
.Molecular mass of substance
density of substance
density of oxygen (Os)
X 32.000 (8.1)
Since 19(51, the Commission on Symbols, Unite and Nomen
clature in Physics has defined the atomic mass unit as onetwelfth
of the mass of an atom of the carbon 1 "2 nuclide. The number
representing the mass of any atom on this scale is called its
atomic mass. (The term "atomic weight" is also used.) On this
scale, which differs only very slightly from the former one, the
mass of the hydrogen atom is nearly 1 amu and the heaviest
known atom has a mass of about 250 amu.
We shall use the symbol .1* to represent, as needed, either
atomic mass or molecular mass.
A quantity of any substance whose mass, in grams, is numeri
cally equal to its molecular mass is called a mole. In the mks
system we define the "kilogram mole" as: I kmole of a substance
is that quantity whose mass in kilograms is numerically equal to
its atomic (or molecular) mass. The mass of 1 kmole of any sub
stance is thus .1 * kg.
The volume occupied by a mole of any gas is called a gram
molecular volume. It is 22.4 liters for a gas at 0°C and 76 cm of
mercury pressure.
The numerical value of Avogadro's number is not easy to
80 Looking In: Atomic and Nuclear Physics
The Atomic Idea
81
measure, and it was not known for some time after Avogadro's
hypothesis was accepted. This constant can be determined inde
pendently from experiments in electrolysis, Brownian motion,
radioactivity, and xray diffraction in crystals. The currently
accepted value of Avogadro's number is
N A = (6.02486 fO.OOOHi) X 10 2S molcculcs/kmolc (8.2)
Example. Compote the number of atoms in a 1.5mg sample of lead,
atomic mass 207.
The mass of 207 atomic muss units (amu) may be thought of as
207 kg/mote. Then
1 5 X I0~ 6 ksr
N = 207ki7kmole X 6 ° 25 X ^" ^m./kmole
= 4.36 X I0 1S atoms
8.3 THE IDEAL GAS LAW
The gaseous state of matter is simplest to analyze, chiefly because
the molecules of a gas arc far apart and do not exert appreciable
forces on each other. The behavior of a gas is expressed by an
equation of state, which relates pressure, temperature, and
volume when the gas is in equilibrium. Numerous empirical equa
tions have been suggested to describe the behavior of gases.
The simplest is
pV  nRT
(8.3)
where p = pressure of gas
V = volume
T = absolute temperature
n = number of moles (or kilomoles) of gas present
In mks units the proportionality constant R is called the gas con
stant per kilomole, and from experiment, it has the value
R = 8.317 X 10 s joules/(kmole){°K) (8.4)
When other units are used for the variables in Eq. (8.3), the gas
constant will be expressed differently; for example,
R = 1.987 cat/ (mole) (°K) = 0.082();> literatm/(mole)(°K)
= 8.317 X 10' ergs/ (mole) (°K)
Xo actual gas obeys Eq. (8.3) precisely at any nonzero pres
sure. Hut this equation holds for all gases when the pressure is
reduced sufficiently. For then the molecules occupy an insignifi
cant fraction of the volume of the container and the widely
separated molecules exert no attracting forces on each other.
It is from these considerations that Eq. (8.3) has importance as
the "idealgas" equation of state.
8.4 KINETIC THEORY OF GASES
Kinetic theory treats atomic and molecular processes and reaction
rates by applying elementary methods of mechanics and statis
tics. We shall examine what the kinetic theory has to say about
the observed properties of a gas: its pressure, volume, and
temperature.
y\
/\
/
Fig. 8.1 Model for kinetic theory of
gas pressure.
West
i
i
i
i
i
i
i
—
East
A
/
We shall consider a gas confined at a fixed temperature in a
cubical container with each side of length L. We make the fol
lowing assumptions:
I.
The molecules have negligible volume; they are "points."
The molecules move in random directions, but every molecule
has the same speed v (obviously an oversimplification, which
we shall reconsider soon).
The molecules exert forces only in collisions.
The collisions with the walls are clastic.
The number of molecules is very large, justifying use of
statistics.
The pressure of the gas may be calculated as the force per
unit area at a wall. Let .V he the total number of molecules in a
cubical container (Tig. 8.1). Then N/8 will be bouncing lietween
the east wall and the west wall. Each molecule in this group
82 Looking In: Atomic and Nuclear Physics
strikes the east wall v/2L times per second. In each clastic impact
the velocity of the molecule changes sign and the change in its
momentum is mv — ( — me) = 2mv, where m is the mass of the
molecule. From Newton's second law, the force on the cast wall
is the total momentum change per second at that wall
„ N i> ,. Nmu %
The pressure is given by
P = A =
J_ Nmjfi _ 1 Nmv*
l a£ "a v
(8.5)
where V = I* is the volume of the container.
To compare this prediction with the idealgas equation we
may rewrite Eq. (8.5) as
pV = i(AW) = \8%m*  %NB t
(8.6)
where E k is the translational kinetic energy of one molecule.
Combining Imjs. (8.3) and (8.0), we have
nRT = fJVA't
(87)
suggesting that the absolute temperature of a gas is proportional
to the kinetic energy of its molecules. Further, since N/n is the
number of molecules per kilomole, that is, Avogadro's number
#,1, we have for the kinetic energy of a molecule
3 R
h\ =  — T = : 'I:T
(8.8)
The constant k, called Boltzmann's constant, is the gas constant
per molecule
k =■ — = 8317 X 10 a j oule s/(kmole )(°K')
A', 6.025 X 1 M molecules/ kmole
= 1.38 X 10" M joule/ (molecule) (°IC) (8.9)
An improvement can be made in our simples) statement of the
kinetic theory by removing the second assumption above, that
all molecules have the same speed. Instead we can say that for
any particular molecule v = v t * + vf + v?. If we have a large
The Atomic Idea
83
number of molecules moving at random, the average values of
iv, tv, and rr are all equal. Equation (8.6) then becomes
pV = llVmi*
ami the v of our earlier discussion is replaced by the "rms ve
locity," the square root of the mean value of the square of the
velocity, VP.
8,5 DISTRIBUTION OF MOLECULAR SPEEDS
In a gas at a given temperature and pressure, we expect that some
molecules will have speeds in excess of the rootmeansquare
value, and others will have smaller speeds. Clerk Maxwell applied
the laws of probability to find the distribution of speeds in a large
number of molecules in a kinetic theory gas. He obtained the
result *
X
4
\Ztt
(&f«
e  mT =/s*r f}„
(8.10)
where N = total number of molecules
N r dv/N  fraction of all molecules whose speeds are between v
and v + dv
T = absolute temperature
fe = Boltzmann constant
m = mass of a molecule
For a gas at any given temperature, the number of molecules
in a speed interval A» increases up to a maximum at the most
probable speed v p of Fig. 8.2a and then decreases toward zero
at high speeds. The distribution is not symmetrical about »„,
for the lowest speed is zero, but the theory predicts no upper
limit for the speed a molecule can attain. For this reason, the
average value v of all speeds is somewhat larger than «„. The
rootmeansquare value v rim is still larger.
As the temperature of a gas is increased, the most probable
speed of the molecules increases in accord with the meaning of
temperature (Eq. 8.7). The range of speeds is greater (Fig. 8.2b);
* L. U. Loeb, "Kinetic Theory of Gases," McGrawHill Book Com
pany, Inc., New York, 1934; Leigh Page, "Introduction to Theoretical
Physics," chap. 9, D. Van Nostrand Co., Inc., Princeton, K.J., 1935.
84 Looking In: Atomic and Nuclear Physics
~kT
/itr
No.
(a)
*/Fs /»=m
200
400
600
800
m/sec
Fig. 8.2 (o) Maxwell distribution of molecular speeds at 0°C.
(b) Maxwell speed distributions at three different temper
atures, T 3 > Ti > T.
there is an increase iti (lie number of molecules which have speeds
greater than a given speed.
8.6 MEAN FREE PATH
TIxe mean free patli is defined as the average distance a molecule
travels between collisions with other molecules. Assume that each
molecule is a sphere of radius R. Consider the motion of a par
ticular molecule among all the other molecules of a gas. It will lilt
any molecule whose center lies within a cylinder of radius 2J9
around its path (Fig. &.',t). In going a distance L, the molecule
sweeps out a volume ir(2R)' 1 L. If there are >i molecules per unit
volume in the gas, the moving molecule will bitx(2R) i Ln molc
The Atomic Idea
85
rules in going a distance L. Its mean free path X is the average
distance per collision:
X =
r(2R)*Ln 4*Rhi
(8.11)
This equation is based on the picture of the single moving
molecule hitting other molecules which are stationary. Actually
the molecule hits moving targets. The collision frequency is
2R
m
Fig. 8.3 Path of a molecule.
increased as a result. More complete analysis shows that the
nieaii free path is reduced to
X =
I
4tt y/2 trn
(8.12)
Example. I" helium gas at 0°C and I atm pressure, the mean free path
of one molecule (or atom, He) is 1.86 X 10 7 m. Estimate the radius of
8 helium atom.
The number of molecules per cubic meter under standard condi
tions m
n =
.V,,
R =
22.4 (m 3 . bnole)
1
= 2.(59 X 10" molceulcs/m'
{Aw y/2 Xh) j
= 1.05 X I0"m
which agrees in order of magnitude with other methods of measure
ment. Note that the mean free path X is about 1.86 X 10 7 m/2
86 Looking In: Atomic and Nuclear Physics
(1.05 X 10 u m) = 900 moleeuliir diameters. From Bq. (S.8) the
speed of the He atoms is
u = = — = 1,310 m/sec
2 m
So the frequency of collision is
v 1,310 m/sec
X 1.86 X 10' m/(2 X 1.05 X 10 10 m)
or 7 billion collisions per Becond.
= 7 X lO'sec  '
8.7 MEASUREMENT OF MOLECULAR SPEEDS
An experimental verification of the. distribution of molecular
speeds predicted by kinetic theory was reported l>y Stern in
(a)
1
m
G&rtn
(4)
^~=©="
Fig. 8.4 Apparatus for measuring molecular speeds, (a)
Stem's rotating drum, (b) Lammert's slotted disks.
i926. Atoms (Ilg) from an oven at known temperature pass
through a slit <S and enter a cylinder C through a narrow slit, in
its wiill (Fiji. 8.4a). With the cylinder stationary, the molecular
beam reaches O diametrically opposite the entrance slit. Hut if
the cylinder is rotated rapidly, the molecular beam is interrupted.
If a point on (he cylinder wall rotates clockwise from O to () v in
the time it takes an Tig atom to cross the diameter d, then the
trace left on the wall by IIg atoms will be displaced counter
clockwise a distance 00% (= 0\0) from the reference point. The
speed of the atoms can be calculated from
Speed of atoms diameter d
Speed of drum surface displacement O a O
The Atomic Idea 87
In Lammert's method, two disks each with ")0 notches were
mounted fi cm apart on a rotating axis (Fig. 8.46), in an arrange
ment similar to that used by I'izeau to measure the speed of
light. For a particular constant speed of rotation of the disks,
only those Hg atoms of a certain speed will be able to pass through
both notched wheels and reach the collector plate P. !3y varying
the speed of rotation and by determining the number of atoms
received at P as a function of their speed, one obtains results in
N(v)
u.o
0,?0
'  \  1
0.15
0.10
^■"1
0.05
90 140
190 240 290
Speed, m/sec
340 390
Fig. 8.5 Speed distribution of mercury vapor molecules at 100°C. (6.
tammert, Zeitschrift her Physik, 56(34): 244253 (1929).]
good agreement with Maxwell's predicted distribution (Fig.
•S..V). (The slight discrepancies with the predicted values, shown
dotted, were attributed to difficulties of alignment.)
8.8 SPECIFIC HEATS
Consider a gas confined at constant volume which is heated. The
specific heat C? of the gas is defined as the heat required to raise
the temperature of a unit quantity of the gas one degree. This
heat is stored in the form of increased kinetic energy of the gas
molecules. From Bq, (8.8) the increase in the internal energy of
1 mole divided by the increase in temperature is given by
(8.13)
88 Looking In: Atomic and Nuclear Physics
Thus our basic kinetic theory makes the challenging prediction
that all gases have the same value of specific heat
C\  §[1.987 calV(moIe)(°C) = 2.98 cal/(mole)(°C)
The value predicted checks well with experimental data for
monatomic gases, but not for gases whose molecules are made up
of two or more atoms.
Toble 8.1 Specific heats (at 15°C)
Type of gas
Gas
C,. (experimental),
cal/{moleH°C)
Monatomic
He
2.98
A
3.00
Diatomic
Hi
4.80
o,
4.96
N,
4.94
CO
4.95
Polyatomic
C0 2
6.74
NH
6.78
C^Hg
9.50
To explain the data of Table 8.1, we may ask whether heating
may result in energy being stored in forms other than transla
tional kinetic energy of molecules, expressible in terms such as
■frnvf. In a dumbbellmodel diatomic molecule (Fig. 8.6b),
there may be kinetic energy of rotation, expressible in terms
such as ^/ur. If the two atoms can vibrate and have a force
constant k, there will be vibrational energy expressible as \l.\r.
Each independent mode of absorbing energy is called a degree of
freedom,/. A theorem of cquipartition of energy, stated by Max
well, says that for a large number of particles which obey New
tonian mechanics, the available energy is equally divided among
the degrees of freedom, .]/.V for each.
Thus modified, our kinetic theory eau be made to agree fairly
well with experimental data for monatomic gases (/ = 8, C„ =
35/2) and for diatomic gases (/ = 5, C = 5B/2). One finds
* The calorie was originally defined as the amount of heat necessary
to raise the temperature of one gram of water one centigrade degree In
1948 it was redefined as 1 caloric = 4.1840 joules. The large calorie
{kcal or Cal) used in nutritional measurements is 1,000 times as large.
The Atomic Idea
89
experimentally (1) that, contrary to kinetic theory, C T varies with
temperature, and (2) that for polyatomic molecules we need to
devise empirical models that differ from gas to gas. We have come
to the limit of validity of classical mechanics when we seek to
describe the behavior of very small particles of matter (molecules
and atoms). Quantum theory is the extension of classical theory
which we need for this {Chaps, I o and Hi). We shall reach another
Fig. 8.6 Degrees of free
dom: independent modes of
energy absorption.
(a) O
(b)
Degrees of freedom
3: trans!.
3 tronsl,
2 rota .
3 trans) .
2 rota .
1 vibra.
(c) OmKD
(d)
limitation in Newtonian mechanics when we deal with particles
which are moving very fast (» — * c). Relativity (Chap. 11) modi
fies Newtonian mechanics in this case.
PROBLEMS
1. Copper which has a specific gravity S.9 has an atomic mass
R3.8 amu. What is the average volume per atom of copper?
Ans. 1.2 X 10" cm 3
2. Compute the rms speed at 0°C of the molecules of («) CO»,
(b) H 2 , (c) Xo. Aits. H02 in sec, 1X4 m '.ee. !()2 m 'sec
3. If the average distance between collisions of CO* molecules under
I atm pressure and at Q°C is ft. 29 X 10 6 in, what, is the time between
collisions? Ans. 1.6 X 10"" sec
90 Looking In: Atomic and Nuclear Physics
■i. In a certain electron microscope, electrons travel 1.0 m from
electron gun to screen. To avoid scattering of electrons by residual
molecules of nitrogen in I lie vacuum chamber, below what pressure
would you recommend operating (lie microscope? The radius of a
nitrogen atom is about 2 X 10 10 m. Ana. p < 8 X 10~ s atm
5. What pressure will 10 gin of helium exert if contained in a 50cm*
cylinder at 2l°C? Would a cylinder rated at 100 atm maximum safe
pressure be safe to hold this helium? 4ns. p « 1,000 atm
Equipped with his five senses, man explores the universe around him and
calls the adventure Science. E. P. Hubble
A series of judgments, revised without ceasing, goes to make up the incon
testable progress of science. DuClaux
The main difference of modern scientific research from that of the middle
ages lies in its collective character, in the fact that every fruitful experiment
is published, every new discovery of relationship explained . . . Scientific
research is a triumph over natural instinct, over that mean instinct which
makes a man keep knowledge to himself and use it slyly to his own advan
tage ... To science this is a crime. H. G. Wells
Wave Motion
To the mathematician the problems of
wave motion offer a field for his
highest power of analysis; to the
physicist they suggest experiments
demanding all the skill at his disposal ;
to the engineer and to those who go
down to the sea in ships these prob
lems are matters of life and death,
while to the poet and the artist they
are "the sea dancing to its own
music " Henry Crew
In the study of wave motion wc arc concerned with the propaga
tion of disturbances in physical systems. A wave is a description
of a disturbance wliicli propagates from one point in a medium
to other points, without causing atiy permanent displacement in
the medium as a whole. Tints sound is a type of wave motion;
wind is not.
Wave motion occurs in a medium in which energy can be
stored in both kinetic and potential form. In an elastic material,
kinetic energy results from inertia and is stored in the motion of
91
92 Looking In: Atomic and Nuclear Physics
the molecules, whereas potential energy results from the displace
ment of molecules against an elastic restoring force. In an electro
magnetic wave, we may regard kinetic energy as stored in the
magnetic field and potential energy in the electric field. In a
traveling wave, one part of the medium disturbs an adjacent
part so that kinetic energy at one point is transferred into poten
tial energy at an adjacent one, and that potential energy becomes
kinetic energy at still another point, and so on.
9.1 TYPES OF WAVES
A wave is a disturbance that moves through a medium in such a
manner that at any point the displacement is a function of time,
while at any instant the displacement at a point is a function of
the position of the point. The medium as a whole does not pro
gress in the direction of motion of the wave. Waves are usually
described mathematically in terms of their amplitude (maximum
displacement from equilibrium) and how the displacement varies
with both space and time. This requires solution of the wave
equation consistent with the boundary conditions for the particu
lar case being studied. In cases most often considered, the wave
equation is a secondorder, linear, partial differential equation.
The general solutions of the wave equation for a onedimensional
space coordinate x are of the form
* = F(x  vt) + G(x + vt)
(0.1)
The functions P and (! are determined by the boundary condi
tions, and the speed v by the properties of the medium. The first
term represents a wave traveling in the positive x direction; the
second term represents a wave traveling in the negative x direc
tion. These are usually sine or cosine waves, for the onedimen
sional case.
A wave moving on a string is an example of a onedimensional
wave. Hippies on water are twodimensional waves. Acoustic and
light waves are threedimensional.
Waves may be classified in accordance with the motion of
individual particles. Transverse waves and longitudinal waves
are the most common types, but there are others. For example,
as a wave moves on the surface of water, the path followed by an
individual particle is either a circle or an ellipse.
Wave Motion
93
Elastic waves, of which acoustic or sound waves are a particu
lar kind, require a medium having two properties, elasticity and
inertia. Elasticity of the medium is needed to provide a force to
restore a displaced particle fo its original position. Inertia is
needed to enable the displaced particle to transfer momentum
to a neighboring particle. In an elastic medium one may have,
in addition to a longitudinal or a transverse wave, a shear wave.
This is a rotational wave which causes an element of the medium
to change its shape without a change of volume.
Light waves, radio waves, and other electromagnetic waves
are not elastic waves and therefore can travel in free space as
well as in transparent media. In a vacuum all electromagnetic
waves travel with constant speed, approximately :j X 10* m/sec.
km
Tvv
n
I » . .
(a)
(6)
Fig. 9.1 Wave fronts; (a) plane; (b) spherical. Arrows represent rays.
In material media the speed is less, and its value depends on the
medium.
Waves may be classified further in terms of time: the perio
dicity or lack of periodicity of the disturbance. If a long coil
spring ("Slinky") is stretched out on a table, a single sidewise
movement at one end will send a pulse or single wave along the
spring. Kaeh particle remains at rest until the puke reaches it,
then moves for a short time, and returns to rest, However, a
continuing toandfro motion applied to the end of the spring will
produce a train of waves. If the motion is periodic, we shall have
a periodic train of waves. An important special case of periodic
wave is a simple harmonic wave in which each particle is given an
acceleration proportional to its displacement and directed toward
the equilibrium position.
94 Looking In: Atomic and Nuclear Physics
An aid in visualizing waves is the idea of a wave front. A wave
front is a surface drawn through points undergoing a similar
disturbance at a given instant. The location of a disturbance
(pulse) at successive e(iial time intervals may be indicated by
drawing successive wave fronts. A line perpendicular to a wave
front, showing the direction of motion of the wave, is called a ray.
Wave fronts spreading from a point source in a homogeneous
medium are spherical. Hut at large distance from the source a
section of the wave front may be treated as practically plane.
9.2 FUNDAMENTAL RELATIONS
A wave is commonly identified in terms of either its wavelength
\ or its frequency J". In any kind of wave motion these two quanti
ses are related to the velocity of propagation > by the simple
equation
/A = v (9.2)
The period is the reciprocal of the frequency. The amplitude A
Wave Motion
95
Medium t
— "Pi A*
Medium 2
(f 2 >u,)
(a) A
(6)
Wavelength
■* — — m Wavelength \j ■* —
Fig. 9.2 When a wave passes from one medium to another, in which the wave
speed is different, (a) the frequency is constant, (fa) the wavelength changes.
is the maximum value attained by the variable of the wave (e.g.,
the displacement) at a given point in space.
The frequency of a wave remains constant under all circum
stances except for a relative motion between the source of the
wave and the observer (see Sec. 9.15, Doppler Effect). The speed
of propagation, however, is dependent on the properties of the
medium (and, sometimes, also on the frequency). Hence the
wavelength will vary with speed in accordance with Eq. (9.2),
as suggested in Fig, 9.2.
9.3 WAVE FORM
A wave form is a pictorial representation of a wave obtained by
plotting the displacement with respect to lime or distance. When
a wave is traveling along a string in the x direction, the shape of
the string at some instant I = can be expressed by an equation
V ■ /(#) when t =
(9,3)
which states that the transverse displacement :/ is some function
f(x) of the distance x along the string. If the wave is moving to the
right with a speed v, the equation of the wave at some later time
I is
y = f{x  Bt)
(9.4)
This gives the same wave shape about the point x = vl at time J
as we observed about point x = at time t = 0.
The relative positioti (displacement) of two points in a wave is
called the phase. Two points which have displacements of the
( =
t=*t
a
b
\e
d(
<vt
Fig, 9.3 A traveling wave.
same magnitude and sign (a and h in Fig. 9.3) arc said to have the
same phase, or to be "in phase." Points c andrf do not have the
same phase, for although both have zero displacement, the dis
placement is decreasing at c, increasing at d. To follow a particu
96 Looking In: Atomic and Nuclear Physics
lar phase in an ongoing wave, wc ask how x changes with t when
x — vt has some particular constant phase value P. Differentia
tion of x — vt = P gives dx/dl = v. So v is the phase velocity
of the wave.
A wave form of considerable importance is one defined by a
sine function
y = f/o sin r (■*•" — i>0 = ij» si" 2»
A
(x t)
(9.5)
The maximum displacement //« is called the amplitude of the
wave. The wavelength A represents the distance between two
points which have the same phase, [■'or a given (, the displacement
// is the same at x, at x + X, at x + 2X, etc. The period T is the
time required for the wave to travel a distance of one wavelength
X, so X = vT, From the second form of LCq. (9,5) it is apparent
that y has the same value at the times t, t + T, i + 2T, etc., at
a given position x.
Wave Motion
97

•
t
T
2
T
Sine wave
litt
y=y sin jr
Square wave
«= 1,3,5
Sawtooth wave
J'0 J'„ r> 1 tint
it = 1,2,3
Fig. 9.4 Some wove forms, defined in terms of sine functions of the
frequency, t T, and its multiples, n/T.
There are many wave forms of interest in physics. To specify
a particular wave Form, one chooses the appropriate function
/(.!■)• It is possible to represent any periodic wave form mathe
matically as a Fourier scries of sine and cosine terms at har
monics (multiples) of the frequency 1, T. Examples are shown in
Fig. 9.4.
9.4 WAVES IN A LIQUID
The waves which we most often see arc those which occur on Ihe
surface of a body of liquid. Waves also occur within a liquid.
Their propagation is made use of in marine equipment such as
the fathometer and Sofar.
Fig. 9.5 Liquid in a channel, showing two positions of o liquid element being
considered.
A quantitative description of a wave motion often can be
obtained by applying fundamental laws of physics to a particular
situation. As an example, consider a channel of unit width and
vertical walls which contains a layer of liquid whose initial depth
is outlined by the solid line in Fig. 9.5. Examine how this can
move to successive positions, as suggested by the dotted line.
We shall apply two physical laws: (1) No liquid disappears or is
created during the process. 2 The rate of increase <.n momentum
of any liquid clement must be equal to the net external force
applied to that element. The force applied to a fluid element at a
cross section such as .1.1 equals the area times the average pres
sure. Since we have assumed unit width for the channel, area =
I X It. The element considered will be the liquid which initially lies
under the solid line of Fig. 9.5, between .4.1 and ('('. After time I,
this same liquid will be under the dotted line between .I'.l' and
98 Looking In: Atomic and Nuclear Physics
C'C. The distance from .1.1 to A' A' is u^l, where h, is the speed of
the liquid lying to tfie left of B.
To satisfy the first requirement mentioned above
(At — hi)x = kiUtt
(9.6)
where x is the distance the liquid originally at rest lias been accel
erated during time interval J.
To satisfy the second requirement
f (A, 2  A.') = "*f '
(9.7)
where ipgki is the average pressure (above atmospheric) in the
liquid of depth Ai, \pghi is the average pressure at CC where the
depth is As, and p is the mass per unit volume of liquid. The left
hand side of Eq. (9.7) is the net horizontal force on the element of
liquid considered. The mass of liquid ph«x is accelerated from
rest to speed a 5 in time t, so (ph«x/t)ui is the rate of change of
momentum.
The two requirements expressed by Eqs. (9.6) and (9.7) now
give for the speed m, of the particles of liquid
, „ (Ai  A;) 2 (Ai + h*)g
2hih a
(9.8)
But the wave speed x/t at which the front of the deeper layer
advances is given by
v= i = 4
(At + A 2 )flig
2A 3
(9.9)
The wave speed v is greater than the speed u of the material
particles.
In this simplified treatment, we have disregarded energy )o>i>.
variation of speed with depth, and a detailed specification of the
shape of the wave front. Vet we have obtained a valid description
of the tidal bores which occur in certain rivers. Such a surge
wave is sometimes employed as a means of dissipating flow
energy at the bottom of a dam spillway. If the channel is so
designed that i/ t = ~r then the velocity of the surge relative to
earth is zero. This form of surge is known us a hydraulic jump.
it can often lie viewed on a small scale by allowing water to flow
Wave Motion
99
from a faucet into a basin. The flow can be adjusted so that in
the basin there can be seen an inner zone consisting of a thin
layer of water moving rapidly outward. Surrounding this is an
outer zone which is a thicker, more slowly moving layer.
The manner in which a continuing oscillatory wave is propa
gated may be examined from considerations similar to those just
suggested for a .urge wave. In shallow water (say, h ,\ < ,'„ . a
2(,
Fig. 9.6 A largeamplitude wove
steepens to form a bore (Fig. 9.5).
wave of small amplitude will be propagated without change of
shape at a speed y/gh, which is consistent with Eq. (9.9),
If, however, the wave height is an appreciable fraction of the
liquid depth, the wave speed is significantly greater at positions
of greater depth. The wave front becomes successively steeper
(Fig. 9.(i), and a bore starts to form.
In deepwater waves, individual fluid particles move in
approximately circular orbits (Fig. 9.7). At the surface, the radius
StiMwoter leve
* — "^ Shallow water
Bottom
(b)
Fig. 9.7 Orbital motion of fluid particles for surface waves (a) in deep water,
and (b) in shallow water.
of the orbit of a particle is equal to the amplitude of the wave.
Hut the radius decreases exponentially with depth, and a region
of almost zero particle motion is soon reached ; hence the behavior
of the wave is unaffected by the total depth of the liquid.
lu shallow water there can tie im vortical motion of particles
at the bottom. The orbits of the particles are ellipses in which the
vertical axis becomes zero at the bottom (Fig. 9.76).
100 Looking in: Atomic and Nuclear Physics
A wave lias equal amounts of potential energy, owing to
particle displacement above or below the stillwater level, and
kinetic energy, owing to the motion of the particles in their orbits.
The speed at which energy is transmitted in the direction of wave
travel is called the group velocity n. us distinct from the phase
velocity v = x/l. In deepwater waves the group velocity is one
half the phase velocity. In shallowwater waves u = e.
9.5 SOUND WAVES IN A GAS
In sound waves usually encountered, the intensity is so small
that the changes in temperature and pressure in the wave are a
very small fraction of the ambient temperature and pressure.
Plane wave Front
«1
Pi
«2
Pi
Fig. 9.8 Plane wave front in a gas.
These waves l ravel at a speed which depends only on the ambient
state of the fluid.
The propagation of a sound wave in three dimensions can be
derived from fundamental physical principles starting in this
way. Imagine, a small prism or a packet of gas enclosed by a
weightless dcformable membrane. The mass within this packet
remains constant. The elasticity is expressed by the idealgas
law. The inertia appears in Newton's second law, from which the
equation for the wave propagation can be derived.
A simpler procedure may be followed in describing the special
case of a plane wave front moving from right to left at constant
speed it, in a gas initially at rest and having density pi. To an
observer moving with this wave front there will appear to be a
steady flow of gas from left to rigbl across the wave Ironi (Kg,
9.8),
Wave Motion
101
Since in a steady flow there can be no accumulation of the
mass at the wave front,
Pi*'i = wis (9.10)
where p« is the density of gas at the right of the wave front and
h 2 is the velocity of this gas relative to the observer moving with
the wave front. Also, an increase in gas momentum across the
wave front requires a drop in pressure from pi to p 2 :
P2>1? — Pl»l 2 = Pl  P'J (9.11)
Tiiis expression is obviously related to Bernoulli's theorem for the
steady flow of on incompressible fluid (p = const). If we consider (he
fluid flowing past two different cross sections of a pipe at different
elevations A] and h~ and apply the principle of conservation of energy,
we get
(9.12)
«{As  Ai) + \ («*•  »,*) = p,  pj
Bernoulli's theorem thus says (hal at any two points along a streamline
in an ideal fluid in steady flow, the sum of the pressure, I he potential
energy per unit volume, and the kinetic energy per unit volume have
the same value.
For a small disturbance where the fractional changes in gas
velocity, density, and pressure are much smaller than unity, these
changes across the wave front can be written as it* = u\ + du,
Pz = p\ + dp, and p : = pi + dp. When we substitute these in
Kqs. (9.10) and (t).ll) and neglect product terms of differential
quantities, we have
Pl du + tt r dp  (9.10a)
2p,u, du + urdp = dp (9.1 In)
By eliminating du from these two equations, we obtain an expres
sion for the wave speed
u  F p
(9.13)
Laplace assumed the compressions and expansions associated
with sound waves should obey the adiabatic gas law, pp—> =
constant where y is the ratio of the specific heats, C p /C r . If this
Relationship for p and p is assumed, the speed of sound becomes
tti = VyRT = J?l
(<ut)
102 Looking In: Atomic and Nuclear Physics
This result, based on the adiabatic law, does not hold for liquids,
for gases at extreme pressures and temperatures, or for acoustic
waves of very high frequencies. However, the pressure fluctua
tions in sound waves range from about 10 9 to 10~ 3 atm, which
justifies the asumption of small disturbance in deriving
Bq. (9.14).
9.6 SHOCK WAVES IN A GAS
In a wave of large amplitude, the wave speed is higher than wi
in regions of condensation (p > pi) and lower than »i in regions
of rarefaction. This causes the wave to distort as it propagates.
Regions of higher condensation overtake those of lower condensa
tion (Fig. 9.9). The thin "characteristic lines" are shown for cor
Fig. 9.9 When wave speed increases
with wave amplitude, the wave form
becomes distorted at successive time
intervals, 1„, I,, 2f i.
f D =
responding points in the wave. The slope di/dx of these lines is
inversely proportional to the speed. The net effect is to steepen
compression regions and to flatten expansion regions. Before the
situation represented at 2d is reached, friction and heattransfer
effects counteract the steepening tendency. The compression part
of the wave propagates without further distortion. It is then a
shock wave.
Bomb blasts start as shock waves, largeamplitude compres
sion waves. Planes traveling at speeds greater than the speed of
sound (Much niiniher = speed of body/ local speed of sound > 1)
generate shock waves which are responsible for the sonic boom
sometimes heard and felt on the ground. When an astronaut
reenters the earth's atmosphere, the early motion of his vehicle
is determined by its shock wave and can be estimated from the
size and velocty of the vehicle and the known temperature,
pressure, and density relations for the wave.
Wave Motion
103
9.7 WAVES IN SOLIDS
Different types of acoustic waves may occur in solids, depending
on the way in which potential energy is stored in the solid.
Transverse waves on flexible stretched strings are described
by an equation of the form
3P" 6\r a
(9. If))
where ;/ is the displacement of the string at a point ,r. The speed
of propagation v is equal to the square root of the ratio of the
tension to the mass per unit length of the string:
* = 4 In
(9.10)
Acoustic waves occur in bars when the bar i> brm and re
leased. Here the restoring force is due to the moment of the forces
about the neutral plane in the bar and depends on the cross
sectional dimensions and on Young's modulus.
Seismic waves which travel through the ground originate from
natural readjustment of the faults in the earth's crust or from
explosions. Both body and surface waves result. The body waves,
which travel through the interior of the earth, may be classified
into dilationai (longitudinal) waves, which are similar to acoustic
waves in compressible fluids, and shear (transverse) waves, which
occur on account of the large shear modulus of most elastic
solids.
From known relationships between propagation speeds and
the mechanical properties of various substances, seismologists
obtain from seismograms valuable information about the strnc
i lire of the earth. Such information can be applied to prospecting
Tor mines and wells.
9.8 ELECTROMAGNETIC WAVES
■lames C. Maxwell recognized about 1801 that the basic equations
for electric and magnetic lields could be combined to give an
equation which resembled the wave equation for mechanical
waves in a fluid {see Sec. Fi.fi).
104 Looking In; Atomic and Nuclear Physics
9.9 SUPERPOSITION OF WAVES
For many kinds of waves, two or more waves ean pass through
the same space independently of one another. One can distinguish
the notes of a particular instrument while listening to a full
orchestra. The displacement of a particle in the medium at any
instant is just the sum of the displacements it would be given by
each wave independently. The principle of superposition states
that the net displacement of a particle is the vector sum of the
displacements the individual waves alone would give it. This
principle holds for an elastic medium whenever the restoring
force is proportional to the deformation. Superposition holds for
Fig. 9.10 Analysis of a complex
wave form.
electromagnetic waves because of the linear relations between
electric and magnetic fields,
The superposition principle does not hold in every ease. It
fails when the equations describing the wave motions are not
linear. An acoustic shock wave has a quadratic wave equation;
superposition does not hold. Hippies which can cross gentle ocean
swells cannot preserve their identity in breakers. Intermodulation
distortion occurs in an electronic amplifier when the system fails
to combine two tones linearly.
An important consequence of the superposition principle is
that it provides a means of analyzing a complicated wave motion
as a combination of simple waves. Joseph Fourier showed that
any smooth periodic function may be represented as the sum of
a number of sine and cosine functions having frequencies which
are multiples of a single basic frequency. The displacement of a
Wave Motion 105
particle in the medium transmitting a complex wave is given by
an equation of the form of
y = A i sin co( + Ai sin 2ut + At sin Swi + ■  •
+ B« + if i cos oil + Bi cos 2U + B,i cos :io>t +
(9.17)
In Fig. 9.10, the wave (dotted) which has an approximately
square wave form is shown to be equivalent to three component
waves with frequencies in the ratio 1 :3:o and amplitudes in the
ratio 1 : : g. The Fourier series representing the square wave is
A A .
y = A sin id + tt sin JW + ■=■ sm out +
9.10 INTENSITY OF A WAVE
(9.18)
In any wave, energy is transmitted through the medium in the
direction in which the wave travels. The amplitude of the wave,
which is the amplitude of vibration of the particles in the medium,
is related to the transmission of energy. Each particle has energy
of vibration which it passes on to the succeeding particles.
In simple harmonic motion, where there is no damping, the
energy of a vibrating particle changes from kinetic to potential
and back, the total energy remaining constant. We may find this
constant energy from an expression for the maximum kinetic
energy
E k  lm(u ui:ix y =
( 2 r ')' "
\m@rfv<>y
2ir % mf 2 y^
(9.19)
where y<> = amplitude of vibration
T — period
/ = frequency
m — mass of the particle
The energy per unit volume in the medium is the energy per parti
cle times the number n of particles per unit volume
~  a2* s 8*/W  iTrppih?
(9.20)
where p = mn is the density.
The intensity / of a wave is defined as the energy transferred
per unit time per unit area normal to the direction of motion of
106 Looking In: Atomic and Nuclear Physics
the wave. The energy tluil travels through such an area per unit
time is that contained in a volume which has unit cross section
and a length equal numerically to the speed 8 of the wave. From
Eq. (9.20)
/ = 2Teh>ppij,r
(9.21)
The intensity is directly proportional to the square of the ampli
tude and to (he square of the frequency of the wave.
When u wave originates at a point source and travels outward
through a uniform medium, at some instant the energy is passing
through the surface of a sphere. A moment later the same energy
is passing through a larger spherical surface. Since the total
energy per unit time is the same at the two surfaces, the intensity
is inversely proportional to the area 4?rr 2 of the surface:
/ =
lirr*
/.":
(9.22)
If instead we have a line source (e.g., a fluorescent lamp), the
energy is spread over successively larger cylindrical surfaces.
The intensity is inversely proportional to the area 27rr/ of the
cylindrical surface:
f =
2jrrt
hi
r
(9.23)
Here the intensity is inversely proportional to the distance r.
For a plane source (e.g., a skylight), which is large compared
to the distance from the source, the energy passes through suc
cessive planes of equal area. There is no divergence of the rays.
In this case the intensity is independent of distance.
As a wave passes through any medium, some energy is ab
sorbed by the medium. Hence the energy >;i<iim through suc
cessive surfaces decreases faster than expected from the change
in area alone. The decrease in intensity due to absorption of
energy is called damping, A wave whose amplitude decreases for
this reason is called a damped wave.
9.11 INTERFERENCE OF WAVES
The physical effect of superposing two or more wave motions is
called interference. Where waves arrive in phase, the interfer
Wave Motion
107
ence is constructive. The amplitude is the sum of the amplitudes
of the individual waves. Where waves arrive 180° or X/2 out of
phase, the interference 1 is destructive.
When two wave trains of different frequency interfere, a series
of alternate maxim;! and minima is produced in the amplitude of
the vibration (l'ig. i).ll). The frequency of these "beats" is the
difference of the two wave frequencies. A familiar example occurs
in sound. If two tones of slightly different frequency are sounded
together, one perceives that the loudness pulsates at the beat
wwvwm i 1 1 1 1 1 1 1
wwwvwm i 1 1 1 1 1 1 1 1
Fig. 9.11 Two waves of different frequency combine to couse beats. Two
coincidences per unit time are shown for wave trains of frequencies 10 and 12.
Frequency. Thus if the tones are middle O (2(i!/sec) and O sharp
(280.5/scc), there will be 16.5 beats sec.
9.12 DIFFRACTION
The bending of a wave around an obstacle is called diffraction.
Diffraction is readily observed as ripples on water bend around a
stick placed in their path.
The principles of diffraction and interference are applied in
the measurement of wavelength of light with an optical diffraction
grating. A transmission grating is a glass plate upon which is
ruled many equally spaced lines, usually several thousands per
centimeter. A parallel beam of monochromatic light falling
normal to this grating (l'ig. 9.12) sends waves in all forward
directions from each slit. Along certain definite directions waves
from adjacent slits are in phase and reinforce each other.
Consider parallel rays making an angle with OB, the normal
to the grating, which are brought to focus at a point P by an
108 Looking In: Atomic and Nuclear Physics
achromatic Ions, If ray AP travels a distance X farther than ray
CP, then waves from .1 and C will interfere constructively at /'
for they differ in phase by a whole number of wavelength*. The
wave front CD makes an angle 8 with the grating. From the small
est right triangle, the path difference X is seen to be CA sin 0.
The distance ('A between corresponding points in the ruling is
called the grating space />. The condition for reinforcement in
the direction 8 is
b sin = X (first order) (9.24)
There are other directions on each side of OH for which waves
from adjacent slits differ by 2X, 3X, IX, etc., and for which the
Fig. 9.12 Diffraction grating.
corresponding bright images P>, /\, etc., are called the second
order, thirdorder, etc., images. The grating equation in more
general form is
b sin 6 = A T X
(9.2.))
where N is the order of the spectrum and b is the grating space.
When white light fulls on the grating, it is dispersed into its
component colors. Spectra are produced at Pi, r\ etc. The dis
persion is greater in the higherorder spectra. In each, the colors
appear in the sequence violet (small X) to red (large X) with in
creasing deviation.
Example. A yellow line and a blue tine of the mercuryarc spectrum h;ivc
wavelengths of 5,791 A and 4,358 A, respectively. In the spectrum
Wave Motion
109
formed by a grating that has 5.000 lines/in., compute the separation of
these two lines in the thirdorder spectrum
o = ttIsL cm = 5.08 X 10" « cm
o.OOO
5,791 A = 5.791 X 10" r ' am 4^68 A = 4.358 X 10 * am
sinfl = ^ = 3(5^?1_X 10 = <n.O ... „. e _ 2();r
" b 5.08 X 10' cm '
sin 8i, =
Separation
3(4,358 X IP" 6 cm)
5.08 X 10 4 cm
'* — ft ~ 5,3
= 0.258 ft, = 15.0°
9.13 STANDING WAVES
Tf a wave on reaching the boundary of a medium is totally
reflected., the reflected wave proceeds in the opposite direction
■fW
(a)
(b)
Fig. 9.13 Standing waves from superposition of waves traveling in opposite
directions; R is the resultant of A and S. The envelope of a standing wave is
shown in (M.
and with equal amplitude (big. 9.13.!. The incident and reflected
waves add according to the principle of superposition. Two such
waves, proceeding to the right and left, may be represented by
the equations
MM)
The resultant may be written
(9.26)
i/ x = yo sin
yi = !/ B sin :
,'/t + t/s — tf » s i n
110 Looking In: Atomic and Nuclear Physics
We may use the trigonometric relation for the sum of the sines
of two angles
sin A + sin II = 2 sin l(A + B) cos %{A  B)
to put Eq. (9.26) in the form
2wx „ I
y = 2y sin y cos 2w ■=,
(9.27)
This is the equation for a standing (no n progressing) wave. A
particle at a particular point % executes simple harmonic motion.
All particles vihrate with the same frequency. But the amplitude
is not the same for all particles; the amplitude varies with the
location x. The points x = «X/2 (where n is an integer), at which
sin (2tx, X) = 0, show no displacement and arc called nodes. The
amplitude has a maximum value 2i/ n at points .<■ =
2« + I X
and
2 2'
such points are called autinodos, or loops (Fig. 9.136).
In general, when a wave reaches a boundary, there is partial
reflection and partial transmission. Consider a stretched string
attached to a second string. When a wave in the first string
reaches the boundary joining the strings, the. reflected wave has
smaller amplitude than that of the incident wave because the
transmitted wave in the second siring carries away some of the,
incident energy. If the second string has a smaller linear density
than the first, reflection occurs without change of phase. If the
second string has a greater linear density than the first, there is a
phase shift of 180° on reflection. From Eq. (9.2fi), it is evident
that the wave travels more slowly in the denser string. From the
relation X  v/f, we conclude that in the denser string the wave
length is shorter. In a study of light waves we frequently observe
this phenomenon of change of speed and wavelength as light
passes from one; medium to another.
9.14 RESONANCE
Free or natural oscillation refers to the oscillation of a body or a
system which has been given a displacement from equilibrium and
then is not acted on by any external or driving force. The body or
system will generally have several distinct frequencies of natural
oscillation.
Wave Motion til
If a system which can oscillate is acted upon by periodic
impulses having a frequency equal or nearly equal to the natural
frequencies of the system, oscillations will occur with relatively
large amplitude. This vigorous response of a system to pulses
nearly synchronous with one of its natural frequencies is called
resonance.
Let us determine the natural frequencies of a stretched string.
When standing waves are established in the string, the end points
will be nodes. There may be other nodes in between. So the
wavelength of the standing waves can have many distinct values.
Since the distance between adjacent nodes is X/2, in a string of
Vib rotor
Fig. 9.14 Standing waves in a string driven at a
frequency nearly equal to a natural frequency,
length t there must be exactly an integral number n of half
wavelengths, X 2, so
X = *
n
7i = 1,2,3,
From Kqs. (9.1) and (9.2ti), the natural frequencies of vibration
are
} 21 yitn/t
n= 1,2,3,
(9.28)
These relations may be demonstrated in a string one end of
which receives energy from a vibrator, such as an electrically
driven tuning fork. The string passes over a pulley, I' in Fig.
i>. 14, and is attached to a weight which maintains the string
under tension I<\ The frequency / of the wave is that of the
vibrator. The wavelength is
9 1 i
/ / V
F_
in I
112 Looking In: Atomic and Nuclear Physics
The wavelength may be varied by changing the tension F, which
changes the wave speed v. Whenever the wavelength becomes
nearly equal to 2l/n, standing waves of large amplitude may be
observed. The string is then vibrating in one of its natural modes
and is in resonance with the vibrator.
Example, What forte must be exerted on the string, using the apparatus
of Kg. 9.14, to produce resonance with the string vibrating in one loop?
The vibration has a frequency 20/sec, the string has a length 18 ft and
weighs o.O ok.
From Eq. (9.28), with n = I,
F = 4 Ifm = 4(18 ft)
400 6.0 slug
sec 2 10 X 32
31 11)
9.15 DOPPLER EFFECT
There is a change in the observed frequency of sound, light, or
other waves caused by motion of the source or of the observer,
\ i ami liar example is the increase in pitch of a train whistle as the
train approaches and a decrease in pitch as the train passes.
In the radar system used for traffic control, the speed of a car is
estimated from the Doppler frequency shift in the radar beam
reflected from the car.
In acoustics the Doppler effect deals with cases of relative mo
tion between listener and source, plus the effect of any motion of
the medium. If the source moves toward a stationary observer
with speed vn, waves emitted with a frequency f s appear to have
their wavelength shortened in the ratio (u — Vn)/u, because of the
crowding of the waves in the direction of motion of the source
(big. 9.15), Theses waves, however, arrive at the listener with the
speed u characteristic of the medium.
If, instead, the listener moves with speed u L toward a sta
tionary source, the waves appear to him to arrive with speed
m + v,l The wavelength in this case is the same as that measured
when both listener and source are at rest in the medium.
Xow consider motion of the medium. Let Vm be the component
of its velocity taken positive in direction from listener to source.
The velocity components v f , and vs are taken to be positive in
the direction from listener to source.
Wave Motion 113
Then the general equation relating the observed frequency
fi and the source frequency fs is
h
f*
u + v L — 9u u + vs — I'M
(9.29)
There an; important differences between the acoustical and
the optical Doppler effects. ( I ) The optical frequency change does
Fig, 9.15 Doppler effect due to motion of the source S toward observer O.
Wave front 1 was emitted when the source was at position 1; wave front 2 Was
emitted when the source was at position 2, etc. The drawing shows positions of
Wave fronts when the source is at S.
not depend on whether it is the source or the observer that is
moving with respect to the other. (2) An optical frequency change
is observed when the source (or observer) moves at right angles to
the line connecting source and observer. No acoustical frequency
shift is observed in the corresponding case. (3) Motion of the
medium through which light waves are propagated docs not
affect the observed frequency.
Analysis of the Doppler effect for electromagnetic waves
114 Looking In: Atomic and Nuclear Physics
(light) requires use of Lorcntz transformations and the relativity
postulate that the wave speed c is the same as measured by all
observers. The result for the observed frequency fa is
Vl  WJ&)
j0 Js 1  (u/c) cos 6„
. cos B s + (v/c)
COS do = . , / ■ } ■; — a
I + {V/C) COS Or
(0.30)
(9.31)
where 0<, is the angle measured in the observer frame and 9s is
the angle that would be measured in the source frame if it were
moving with velocity v relative to the observer frame.
The term transverse Dtrppler effect refers to the relativistic,
directionindependent factor in the equations above, fo m
Is \/l — (u'/c 1 ) This shows that the observed frequency will be
less than the source frequency regardless of the apparent direction
of motion of the source.
The radial Doppler effect is the directiondependent factor
and, like the acoustical Doppler effect, is understandable on the
basis of classical physics, fo « /s/[l — if>/c) cos 0«.
In MK18 II. ]•'„ Ives and (1. It. Stilwcll measured frequencies in
the spectrum emitted by moving hydrogen atoms and compared
the frequency shifts with those predicted by the equations above
for the transverse Doppler effect. This became an experimental
verification of the special theory of relativity and of the "dilata
tion of time" (Sec. 14.!)).
QUESTIONS AND PROBLEMS
!. Knergv can lie transferred by particles as well as by waves. How
can you distinguish experimentally between these methods of energy
transfer?
2. When waves interfere, is there a loss of energy? Kxplain.
3. Why don't wo observe interference effects between the light
beams omitted from two flashlights, or between the sound waves from
violins in an orchestra?
4. A line source (fluorescent lamp) emits a cylindrical expanding
wave. Assuming the medium absorbs no energy, find how the amplitude
and intensity of the wave depend on the distance from the source.
5. A cord 75 cm long has a mass of 0.252 gm. It is stretched by a
load of 2.0 kg. What is the speed of a transverse wave in this cord?
Am. 242 m/scc
Wave Motion
115
6. find the speed of a compressional wave in a steel rail whoso
density is 490 lb/ ft 3 and for which Young's modulus has a value 29 X 10 6
lb/in.*. Am. 5,200 ft/scc
7. Compute the speed of sound waves in air at 0°C. The average
molecular weight of air is 29, y = 1. 40. and R = 8.3 X 10 a joules/
(kmole)CK).
8. If a person inhales hydrogen and then speaks, how will the
characteristics of his voice be changed? How would the situation be
changed if carbon dioxide were used?
9. A student places a small sodium vapor lamp just in front of a
blackboard. Standing 20.0 ft away, he views the light at right angle; to
(be blackboard while holding in front of his eye a transmission grating
ruled with 14,500 lines, in. He has his assistant mark on the hoard the
positions of the first order diffracted images on each side of the lamp.
The distance between these marks is found to he 14 ft, 2 in. Compute
the wavelength of the light.
An ocean traveler has even more vividly the impression that the ocean is
made of waves than that it is made of water. A. S. Eddington
False facts are highly injurious to the progress of science, for they often
endure long; but false views, if supported by some evidence, do little harm,
for every one takes a salutary pleasure in proving their falseness.
C. R. Darwin
What art was to the ancient world, science is to the modern.
Benjamin Disraeli
Science and art belong to the whole world, and the barriers of nationality
vanish before them. Goethe
*
Electric and Magnetic Forces
117
10
Electric
and Magnetic Forces
1 cannot help thinking while I dwell upon them that
this discovery of magnetelectricity (induction) is thB
greatest experimental result ever obtained by an in
vestigator, J. Tyndall
Electric charges and electric and magnetic forces are important
in many experiments? designed to reveal the structure and be
havior of atoms. All visual information cornea to us in electro
magnetic waves, and study of the ultimate structure of atomic
nuclei depends on electromagnetic processes and detectors. We
shall outline here only the main ideas in electricity and magne
tism needed for our study of atomic and nuclear physics.
The study of electricity dates from the observation ((500 ux.)
that bits of straw and other materials arc attracted to rubbed
amber. The study of magnetism dates back at least as far, to the
observation that magnetite stones attract iron (but not other
substances, lieneralh , These two sciences were developed sepa
rately until 1820, when Hans Christian Oersted observed a rela
tion between them: An electric current in a wire can affect a
magnetic compass needle. However, the fact that electricity and
116
magnetism were initially developed as separate sciences has led
to some inconveniences in concepts and units which the viewpoint
of the inks units (which we shall use) seeks to minimize.
10.1 CHARGE AND MATTER
We anticipate experimental evidence described in later chapters
to summarize, some modern basic knowledge. Experiments on
the electrification or charging of bodies show that there are two
Nucleons:
Neutrons O O
Protons © *
Nuclei
Atoms
:h©(§)i
Mo I ecu
Electrons 9©
'"(©0©)
Compounds I ( }p )
 — ** le
Visible matter
Fig. 10.1 Composition of matter.
kinds of charge. A glass rod may be rubbed with silk, placed in a
stirrup, and suspended horizontally on a silk thread. If a second
ulass rod is also rubbed with silk and then brought near the
rubbed end of the first rod, the two rods will repel each other.
Hut a hardrubber rod electrified by rubbing with fur will attract
the glass ra cJ. Two rubber rods rubbed with fur will repel each
other. The charges on the glass and hard rubber must be different.
We add the following details to the atomic picture of Chap. 8.
An atom has most of its mass concentrated in a very tiny (10 13
cm) nucleus. The simplest atom, hydrogen, has a nucleus which
comprises a single proton. All other nuclei contain, in addition
to protons, one or more neutrons. Each atom has circulating
118 Looking In: Atomic and Nuclear Physics
around its nucleus a number of electrons equal to the number of
protons within the nucleus. The mass of the electron is about
1/1,840 the mass of a proton or neutron (Table 10.1). An arbi
trary convention adopted in Franklin's time for the sign of the
two kinds of electric charge leads us to call the electron charge
negative, the proton charge positive. A neutron lias zero charge.
Toble 10.1
Properties
of some baste particles
Particle
Symbol
Charge
Mass, kg
Electron
Proton
Neutron
P
n
e (= 5.60 X 10"coul]
9.108 X 10 31
1.672 X 10""
1.675 X 10 «
An element may be designated by symbols zEl 1 , such as «Be*
for berillium. The atomic: number Z represents the number of pro
tons (or electrons) in the atom. Its mass number A represents
the number of nucleons (neutrons and protons) in the nucleus.
The number of neutrons is .1 — Z,
The chemical properties of an atom are determined by its
atomic number. Two atoms which have the same atomic number,
but whose nuclei contain different numbers of neutrons, are said
to be isotopes of the given element.
Objects can be electrified, or charged, either positively or
negatively by the removal or addition of electrons.
Charges of like sign repel ; unlike charges attract.
In the atomic model proposed by Niels Bohr in 1913, elec
trons arc pictured as whirling about the nucleus in circular or
elliptical orbits. The centripetal force needed To hold an elect nm
in its orbit is provided by the force of attraction exerted by the
positive nucleus on the negative electron.
In addition to the electrostatic (coulomb) forces between
charges, there are forces which depend on the relative motion of
the charges. These forces determine the magnetic behavior of
matter.
10.2 COULOMB'S LAW
Coulomb's law (1785) expresses the experimental observation
that the force of attraction (or repulsion) exerted by one charged
Electric and Magnetic Forces 119
object on another is proportional to the product of the charges,
q x and q 2 , and inversely proportional to the square of the distance
r between them (where the objects are regarded as "point"
masses) :
F = k
<?><7s
(10.1)
The proportionality constant k is a positive number whose value
depends on the system of units.
Tn the electrostatic system of units (esu), the unit of charge
is defined conveniently to make k = 1 in Eq. (10.1): One stat
coulomb is that quantity of charge which repels a like charge
with a force of one dyne when the charges are spaced one centi
meter apart in a vacuum (or practically, in air).
However, the meterkilogramsecond (mks) system of units
defines a unit for current (ampere) as a fundamental unit; the
unit for charge (coulomb) becomes a derived unit. The ampere
is defined in terms of an electromagnetic experiment. The ampere
is the strength of that constant current which, maintained in two
parallel, straight, and very long conductors of negligible cross
section placed in a vacuum at a distance of one meter from each
other, produces between these conductors a force of 2 X 10 "'
newtou per meter of their length. The coulomb is that charge
transferred by an unvarying current of one ampere in one second.
In principle, we have only to measure the force, in newtous,
between two 1coul charges separated by 1 m in vacuum to hud
k in mks units. The experimental value is
k = 8.987 X 10° « 9 X 10* newtonm'/coul 2
(10.2)
In the socalled rationalized mks system of units, a different
constant t , called the permittivity of free space, is introduced in
the equation for Coulomb's law
/■' 
so that
eo =
4ttc r 2
(10.3)
1
4irfc
4tt(8.987 X JO 11 ) newtonmVcoul 2
= 8.85 X 10 ' coul7newtonm 2 (10.4)
120 Looking In: Atomic and Nuclear Physics
The arbitrary inclusion of the Factor 4ir in Coulomb's fundamental
law makes certain derived formulas more convenient. No it's then
appear in formulas referring to plane surfaces, a factor 2tt appears
in "cylindrical" formulas, and 4jt appears in formulas relating to
spherical symmetry. For example, Table 10.2 gives expressions
for the capacitance C (charge held per unit potential difference)
for capacitors of different symmetry as expressed in unratioual
ized units and in rationalized mks units.
The vehemence with which questions of units have long been
argued may be inferred from Oliver Ileaviside's statement (1893)
Table 10.2 Comparison of expressions for capacitonce
Unrationalized
units
■ i </ — ^ Plane capacitor
C =
Awd
Rationalized
units
C = iAJd
L
i i Coaxial cylinders
ji (I
2 In b;a
C=
2 In b/a
Concentric spheres C= —
Ah
b a
„_ i (4r)ab
b a
that "the unnatural suppression of Aw in the funrationalizedj
formula for central force, where it has the right to he, drives it
into the blood, there to multiply itself, and afterward to break
out all over the body of electromagnetic theory."
The mks units were adopted by international agreement for
scientific and engineering use beginning in 1940, but actual
acceptance of the mks system has progressed slowly. We shall use
the rationalized mks system of units.
The statement that Coulomb's law applies to "point" charges
means, practically, that charges qy and q> must be associated with
bodies whose dimensions are negligibly small compared to r. The
evaluation of the constant k above holds only for the case where
the two charged particles are in vacuum. If they are immersed in
some medium, say, oil, the polarization of its molecules greatly
Electric and Magnetic Forces 121
diminishes the force between charges r/i and q*. Coulomb's law is
then written F — <7ifj a /4jrer s , where « is replaced by the larger
number e, the permittivity of the material in question.
10.3 ELECTRIC FIELD INTENSITY
If there are several charges Q u Q>, Q 3 , . . . , in fixed positions,
and we bring up another charge 7, it will experience a force. We
say that the fixed charges set up an electrostatic field about them
and the charge q experiences a force when in this field. We define
the electric field intensity as the net force per unit + charge
R
force
charge
+q
(10.3)
Electric field intensity is a vector quantity. Its mks units are
newtons per coulomb (or volts per meter, from Sec. 10.4). We
can often calculate the value of K at each point of a region of
space; these values determine the force on (and hence the motion
of) a charged particle in that region.
F = gE
(10.(>)
The electric field near an isolated point charge Q is easily
calculated. If a test charge q is brought to a distance r from Q, it
experiences a coulomb force
F = q ®
~ (4x eo )r 2
The magnitude of the field is then
P _ Q
K 
+q (4areo)r*
(10.7)
(10.8)
This electric field intensity is represented by a vector which, at
each point in space, points directly away from Q if Q is positive
or directly toward Q if Q is negative.
Example. Two charges, q, = —75 X 10 9 coul and q t = + 75 X 10 9
t'tiLil, are 8.0 cm apart in uir. Find I he electric field intensity E at a
122 Looking In: Atomic and Nuclear Physics
point P, which is 5.0 era from each charge (Fig. 10.2). The field intensity
due to 171 U represented by the vector PA and is given by
2?, _«L
4weor i
= 75 X 10'(9 .0 X 10') newton
(0.05)* ioul
= 27 X 10* newton/coul
The field Ei due to charge q* is also 27 X 10* newton/coul in magnitude,
but its direction is that represented by vector PC. The resultant field E
9 l =75xT0" 8 ?,=> +75xicr D
Fig. 10.2
is represented by vector PR. Since triangles TKiP and PRC are similar,
one may write the following proportion:
PB = DG or _B 8.0 cm
PC PG 27 X 10* newton/coul 5.0 cm
E = 43 X 10* newton/coul, parallel to the tine joining q t and q it
10.4 ELECTRIC POTENTIAL
In electrical phenomena the concept that is important in cases of
energy transfer is that of potential difference. If we move a charge
through an electric field, we exert a force through a distance, and
so do work. The force exerted at each point is proportional to the
amount of charge moved, and thus the total work is proportional
in this charge. The electric potential difference between positions
b and a is the work done per unit + charge in carrying charge
from a to b
AV= V„  V a = ^ (10.9)
Potential is measured in volts: 1 volt = 1 joule/caul
Electric and Magnetic Forces 123
Considering a field due solely to a fixed charge Q, we shah
compute the work done by an external agent in bringing another
charge q from a great distance (infinity) in to P at a distance
H from Q. As q is moved an infinitesimal distance ds along an
arbitrary path (Fig. 10. .'5), it will be acted on by a practically
Fig. 10.3 Calculation of potential at a point P.
constant force F = qQ/4irt D r s . The work done on the system of
charges by the agent exerting force F is
dW = F cos 6 ds = 7^, dr
(10.10)
where the negative sign comes from the fact that F and ds
cos fl (= dr) are vector quantities in opposite directions. The
total work done in bringing q from =o to R is then
IK = ~ ( lQ f R< t= flQ
4ir«o J" r s hrtjt
(10.11)
Dividing by q in Eq. (10.1 1), we have the work per unit charge,
which is the potential, given by
v R  r„ =
Q
4aW£
(10.12}
Strictly, we have defined only difference of potential. Jf arbi
trarily the potential is taken to be zero at infinite separation of
the charges (l'» = 0), then the potential (sometimes called the
124 Looking En: Atomic and Nuclear Physics
"absolute" potential^ at a distance r from an isolated point charge
Qis
Q
V 
4xe r
(10.13)
Tf q and Q are both positive (or both negative) charges, then
the external agent bringing </ toward Q exerts a force in the direc
tion of the motion and does a positive amount, of work on the
system. II" k(ijQ '"). This work is stored in the system as poten
tial energy
qQ
P =
4vt r
= qV
(10.14)
Tin's potential energy can be recovered. If the charge q at a point
distant r from Q is released, it will fly off; its potential energy is
converted into kinetic energy.
If q and Q have different signs, then in the trip from «; to It,
the agent will have to hold q back (to prevent acceleration).
The IF of Eq. (10.1 1) will be negative. Because energy is trans
ferred from the electric field to the agent, the charges are placed
in a configuration of lower potential. Energy would have to be
put back into the system to separate the charges again to infinity.
These ideas will be used in calculating the energy stored in an
atom of hydrogen, where a positive nucleus attracts the negative
electron (('hap. lo).
lisniiifih. Kleetrons which leave a healed filament with negligible energy
arc accelerated to pass through an aperture in a metal plate maintained
at. a potential of !)00 volts above thai of the filament. What is the final
s))eod. of the electrons?
Each electron has a charge of — 1 .130 X 10"" coul and a mass of
9.00 X 10" 11 kg. The electron gains kinetic energy equal to the work
dime on it in falling through potential difference V.
(l)»w* = Ve
Hence
">/?(
2 X 1.6 X lO" 1 ' coul 901)
9.11 X 10" kg
*)'= M X
I0*m/sec
The kinetic energy attained by an electron in falling through a
potential difference of 1 volt is given the name electron volt (ev).
1 ev  (e)(AV) = 1.602 X 10 ,u coul (1.00 joule/conl)
= 1.002 X 10 '• joule
Electric and Magnetic Forces 125
10.5 ELECTROSTATIC DEFLECTION
The deflection of charged particles by electric and magnetic fields
has been important in the identification of elementary particles
and in the development of such useful devices as the cathoderay
oscilloscope and the mass spectrograph.
As a special case, consider a parallelplate? capacitor (Fig.
10.4) with charge —Q on the upper plate and +Q on the lower
Fig. 10.4 £ = AV, s for o uniform
electric field.
z
I
+ Q
plate. If the distance s between the plates is small compared with
the other dimensions, the electric field E is uniform in the region
between the plates. If we lake a small charge </ from the upper
plate across to the lower, the work done is the product of the con
stant force liq and the distance s. From the definition of potential
difference, the work is also the product" of the charge q moved and
AV. By equating these, A'r/s = q AV, we have
E 
AV
for uniform field
(10.15)
A device for studying the charge and mass of particles con
sists of an evacuated tube in which a narrow beam of particles,
defined by slits c, and <•■. passes between the plates of a parallel
plate capacitor and then impinges on a fluorescent screen S' where
it produces a visible spot (Fig. 10.;">), The x component, of the
velocity of a particle suffers no change as the particle passes
through the capacitor and goes to the screen. In the electric field
P
1
—TT
AV
+
kL
4—
S'
Fig. 10.5 Electrostatic deflection of a beam of charged particles.
126 Looking In: Atomic and Nuclear Physics
of the capacitor, a positive particle will experience an upward
acceleration n tf from tlie force exerted by the field E on the charge
F v _ qE
m m
(I O.K.)
The particle emerges at the right side of the capacitor with
velocity components
v x = v
I
m m v
(10.17)
where ( = l/v, the time required for the particle to pass through
the capacitor.
The particle emerges from the capacitor at an angle 8 with its
original path, where
. ? T.' I
ten b . "JL = ■' E
v x m
(10.18)
The deflection A C observed on the screen is the sum of the deflec
tion AB which the particle incurs while in the capacitor and
deflection BC brought about by the v„ velocity component while
the particle travels distance D.
2 m ir
BC = Dt&n0 = ^E,D
(10.19)
(10.20)
Thus the measured deflection AC is as if the beam were abruptly
deflected through an angle midway through the capacitor:
AC = AB + BC = \ q —,+ (>E {  (d + A tan 8 (10.21)
2 mv* mo \ 2/ v '
If we measure AC, D, and I, we can find tan 6. Measurement
of AV and $ determines E and, through Eq. (10.18), gives a value
for q/mv 1 . If we know the initial speed v of the particles, we can
iind a value for q/m, or vice versa. The experiment does not
determine q and m separately. The experiment is usually done in
such a way that is a very small angle, so that tan = (ex
pressed in radians). Then, for a given instrument, is inversely
Electric and Magnetic Forces 127
proportional to the kinetic energy of the particle. Such deflection
experiments are important in identifying, sorting, and utilizing
charged particles.
Toble 10.3 Chargemoss ratios for
several particles
Particle
q/m, coul kg
Electron
Proton
« particle
•1.75V X I0 11
9.579 X 10 7
4.822 X 10 7
10.6 ELECTRIC CURRENT
Electric charges in motion constitute an electric current. In
metallic conductors there are many "free" electrons, that is,
electrons not bound strongly to particular atoms of the metal.
Each electron moves in an irregular path, continually colliding
with atoms of the metal. If a wire is connected across a battery,
an electric field is set up within the metal. The electrons tend to
drift from regions of low potential to regions of high potential.
This electron "wind" is the current. The continual collisions are
responsible for the resistance of the metal. The kinetic energy
gained by the electrons from the field and given up in collisions
is l he power loss iR which produces heating in any current
carrying conductor.
In electrolytic solutions, in some types of vacuum tubes, and
in certain solidstate devices, electric current may result from the
Positive plate +. — Negative plate
t
,/) Ammeter
Electron flow
High potential
Conventlal current
Low potential
Fig. 10.6 Direction of electron flow and of con
ventional current.
128 Looking In: Atomic and Nuclear Physics
motion of both positive and negative charges. Any currenl direc
tion is a convention; the choice of sign is arbitrary. In this honk
we shall regard the direction of conventional current as that of
the flow of positive electricity. The conventional current is from
high potential to low potential ("from + to — "} in the external
circuit.
If two parallel wires carrying current in the same direction are
brought near each other, they attract each other. This effect is
the basis for the definition of the ampere, the inks unit of current
(Sec. 10.2). This attraction is not an electrostatic (coulomb) force
between unbalanced charges. It is a magnetic force arising from
the motion of charges. It is convenient to discuss these magnetic
forces in terms of a field.
10.7 MAGNETIC INDUCTION
The basic vector for describing a magnetic field is called the mag
netic induction, B. (Magnetic field strength would be an appro
priate name for B, but historically this name has been assigned
to another vector H connected with magnetic fields.) We identi
fied and measured an electrostatic field in terms of the force
exerted on a unit positive charge. Kxperimetitally we identify the
presence of a magnetic field from the fact that if a magnetic field
is present, a moving electric charge will experience a sideways
magnetic force.
The magnetic induction H is defined as the vector which satis
fies the relation
F = q(v X It)
(10.22)
where force F, charge q, and velocity v are the measured quanti
ties. This notation of a "vector cross product" means that F is
Fig. t0.7 F = q(v X B)
Electric and Magnetic Forces 129
perpendicular to both v and B and directed so that, if the fingers
of the right hand are directed from the direction of v (around
through an angle of less than 180°) to that of B, the right thumb
will point in the direction of F. The magnitude of F is given by
F^qvIixmB (10.23)
where 8 is the angle included between the positive directions of
v and B, Xote that a vector cross product v x B is zero if v is
parallel or antiparallcl to B, Notice also that A x B = — B x A;
that is, A X B is equal in magnitude but opposite in direction
to B X A.
The unit for B, from Eq. (10.22) is newton/(coulomb meter/
second). This inks unit for B is given the name weber/meter 1
we her _ newton
m 2 eoul m/sec
 1
newton
ampm
An earlier cgs unit for H, still often used, is the gauss.
weber
1
nv
= 10* gauss
The summation of B over a surface J'B ■ r/s is called the magnetic
flux *. The weber is the unit of flux.
When a charged particle moves through a region in which
both electric and magnetic fields are present, the resultant force
on the particle is given by
F = ryE + f/(v X B) (10.24)
Only in the special case where E, v, and B are suitably oriented
can we replace Eq. (10.24) by a scalar equation which suggests
straight, addition:
F = q E + qv B sin B (10.25)
where is the angle included between the positive directions of
v and B.
10.8 MAGNETIC FORCE ON A CURRENT
An electric current may be visualized as moving charges. Assume
that in a conductor of length I there are n conduction electrons
per unit volume, each with charge q and each having an average
drift speed i\ (The negative electrons drift in a direction opposite
130 Looking In: Atomic and Nuclear Physics
to that of the current, Fig. 10.8.) The distance an electron moves
per second is v. The volume of charge passing a certain cross sec
tion A of the wire is Av. In this volume there are nAv = JV con
duction electrons. If this conductor is in a uniform magnetic
/
U
5
Fig. 10.8 Representation of an
electric current.
induction B, the force q(v xB) on each moving charge in the
conductor produces a force on the conductor of length / which is
F = Nq(v X B)
But the velocity is 1//, and Nq/t is the current i, so the equation
for the force hecomes
F = Nq (j X b) = ( N f\ (1 X B) = »( X B) (10.26)
or
F = Bil sin d
10.9 MAGNETIC DEFLECTION OF CHARGED PARTICLES
The force on a charged particle moving in a magnetic induction
is at right angles to B and to v. The particle is accelerated, I nit
always perpendicular to its velocity. The magnetic force changes
the velocity (vector) but not the speed (scalar). No work is done
on the particle by the magnetic force, for cos 9 = in the expres
sion W = /F cos da.
When a charged particle enters a uniform held with its ve
locity perpendicular to B, the particle experiences an acceleration
of constant magnitude qvB/m perpendicular to its velocity. The
particle describes a circular path with constant speed. The cen
Fig. 10.9 Path of charged particle in plane
normal to 1i.
Electric and Magnetic Forces 131
tripetal force needed to keep the particle in circular motion is
supplied by the magnetic side thrust. Since the velocity of the
particle is always perpendicular to the induction, sin 8 = 1 in
Eq. (10.25). Newton's law F = ma then can be expressed as
The momentum of the particle mv can be found if we know B
and q and measure r. If measurements of electrostatic deflection
and magnetic deflection are carried out on the same beam of
charged particles, one can determine both q/m and v for the
particles. In this way Thomson measured the charge/ mass ratio
for electrons in 1897 (Chap. 11). Similar deflection methods arc
used today in some types of mass spectrometers to obtain accu
rate values of q/m for ions and isotopes.
10.10 MAGNETIC INDUCTION OF A CURRENT
We have just considered problems relating to the forces exerted
by a magnetic induction on a moving charge or on a current
carrying conductor. A second class of problems involving mag
netic lields concerns the production of a magnetic induction by a
currentcarrying conductor or by moving charges.
The relationship between current i and magnetic induction B
is given by Ampere's law. The magnetic induction at a point P
arising from a current i in a wire is the vector sum of contribu
tions from every element of the wire. The induction at P due to
the current in element d\ of the wire is
d\\  I \ _ til X r
Air r'
(10.27)
We have again used the notation of the vector cross product. The
magnitude of rfB is given by
dB 
idl
4w r 2
u.,
sin 8
(10.28)
The constant u,. is called the permeability of free space. In the
mks system its value is
/in = 4jr X 10 7 joule/amp*m
132 Looking In : Atomic and Nuclear Physics
The direction of r/B is perpendicular to the plane of the vectors
dl and r, and such that if the fingers of the right hand arc turned
(through an angle less than 180°) from dl to r, then the right
thumb will point in the direction of dli.
Fig. 10.10 Induction dU contributed by cur
rent element i dl.
Ampere's law as expressed in the last two equations cannot
be subject to direct experimental check, for we cannot isolate an
element i dl of an electric circuit. Actually Ampere's law was not
deduced from any single experiment. Rather it summarizes many
experiments dealing with magnetic effects of circuits of different
geometry and witli magnetic forces exerted by currents on each
other.
To illustrate the tise of Ampere's law we shall calculate the
magnetic induction (I) at the center of a circular loop, and (2)
at a point near a long straight conductor. At the center of the
loop (Fig. 10.11) the direction of the magnetic induction H is
Fig. 10.11 Induction B (out of page) at
center of a circular loop.
Electric and Magnetic Forces 133
perpendicular to the plane of the current elements i dl and r, in
the sense given by the cross product d\ x r, or out of the page.
Since a radius to any point on the loop is perpendicular to the
tangent to the circle at that point, sin = 1 in Bq. (10.28).
Writing r d<p for dl, the magnitude of B at the center of the loop is
B = toL* r*"= ■>,■ 0&»>
To calculate the magnetic induction at a distance R from an
infinitely long straight wire (Fig. 10.12), it is convenient to take
Fig. 10.12 Induction B near a Mroight conductor.
the z axis along (he wire in the direction of / and with the origin
O at the point of the wire closest to l\ The upper and lower halves
of the wire make equal contributions so we may compute B by
integration of the expression
B = 2 f u i
/*.
sin
'/:
4x }  » r
For the section of wire below the origin (z < 0) we have
(10.30)
R
— tan 8
and
R
— sin 6
dz = d(.^\=«4
\tm 0} sin 2
de
I
Equation (10.30) may be written in terms of one variable 0:
R _ tint [*ft . . sin 2 e R d0 noi r*te .
134 Looking In: Atomic and Nuclear Physics
giving
B =
li.nl
2xii
(10.32']
10,11 DIRECTION RULE
The relative directions of the vector quantities in Eqs. (10.22)
and (10.27) implicit in the vector cross products can be remem
bered conveniently from the following rules:
(a) If, in imagination, the right hand grasps a current
=^z^
(a)
(M
Fig. 10.13 Magnetic induction B near an electric
current: {a} Side view, i toward right; (b) end view,
j out. It counterclockwise.
Current "in"
Current "out"
B
B,
©
(a)
(b)
(c)
Fig. 10,14 Force on a currentcarrying conductor is from strong field region
toward weak field region.
carrying conductor with the thumb pointing in the direction of
the conventional (+) current, the fingers encircle the wire in the
same sense as the magnetic induction B. Thus in rig. 10.13a, B
is out of the page above the wire (indicated bj dots) ; B is into
the page below the wire (indicated by crosses).
Electric and Magnetic Forces 135
(h) When a currentcarrying wire is in a magnetic field, the
magnetic force on the wire is directed from the region of stronger
induction toward the region of weaker induction.
Consider a wire perpendicular to the page carrying a current
into the page (Fig. 10.14a). The local magnetic induction encir
cles the current in the clockwise sense. If now this currentcarry
ing conductor is placed in an external field B, (Fig. 10. 1 4b),
B and B\ reinforce each other above the wire (strong field region)
and partly cancel each other below the wire (weak field region).
The force F on the wire is down. A representation of the net field
due to B and Bj is shown in Fig. 10.14c,
10.12 INDUCED CURRENTS
A further important relation between magnetic fields and electric
current is the principle of induced emfs on which the design of
generators, transformers, and motors is based. Michael Faraday
and Joseph Henry, at about the same time (1831), showed that
an emf is induced in a conductor when there is any change of mag
netic flux linked by the conductor. It is convenient to consider
this single principle from two viewpoints, (i) An emf e is induced
whenever a conductor moves across a magnetic field
e = IvB sin d
(10.33)
where I = length of wire
v = its velocity
— angle between » and B
When mks units are used on the right of Eq. (10.33), e is given
in joules per coulomb or in volts.
(2) An emf is induced whenever the flux ($ = BA) changes
through a circuit:
e = —
d*
dt
(10.34)
I f the rate of change of flux is in webers per second, the emf e is
in volts.
Lena's law states that whenever an emf is induced, the induced
current is in such a direction as to oppose (by its magnetic effects)
136 Looking In: Atomic and Nuclear Physics
the change inducing the current. Lena's law is really an example
of the conservation of energy principle.
As applied to Fig. 10. l. r >, Lea* 's law says that the direction of
current induced in the moving wire must he Mich as to oppose its
motion. This requires the magnetic force on the wire to be toward
Fig. t0,!5 Emf induced in wire ob
moving across a uniform magnetic
field.
the left. From Sec. 10.1 1 b, the net induction ahead of the wire (at
right) must be greater than that behind the wire (at left). To
reinforce the external li, directed out, ahead of the wire, the
induced current must be down, from a to b, from tint righthand
rule (Sec. 10.1 In).
a
II
b
Fig. 10,16 Emf induced in a coil moving
from a to b in nonuniform magnetic field.
Note that in this "generator" b is at a higher potential than a.
Positive charge is forced to flow from a (low potential) to b (high
potential) by the work an external agent does in moving the wire
ab against the magnetic force. In the external circuit bca, the
charge flows from high potential toward low potential ("from
+ to — "); it can do useful work, and it produces ,/>' heating in
the conductor.
Electric and Magnetic Forces 137
QUESTIONS AND PROBLEMS
1. State some similarities and some differences between the phe
nomena of electric fields and gravitational fields.
2. A positively charged rod is brought near a ball suspended by a
silk thread. The hall is attracted by the rod. Does this indicate that the
ball has a negative charge? Justify your answer. Would an observed
force of repulsion be a more conclusive proof of the nature of the charge
on the ball? Why?
3. The circuit in the diagram consists of two concentric circular sec
tions AC and l)E and two radial sections CI) and HA. There is a cur
Fig. 10.17
rent i in the direction shown. Starting from Kq, (I0.2S). derive an
expression for the flux density at O, the common center of the arcs.
Show the limits of integration arid any special values of factors in die
equation. Show clearly the contribution of each part of the circuit.
What is the direction of the (lux density at 0? Ans. ? ol(fii ~ _ ^ij
4. A 5.0m straight wire ab (Fig. 10.18} is allowed to fall through a
uniform magnetic induction of 2.0 XII) 5 webcr/m* directed pcrpen
Fig. 10.18
x
a
V
dicular to the wire, (n) What is the emf induced in the wire at the
instant its speed is 3.6 m/see? (6) What is lire direction of this emf in
the wire? (c) Which end of the wire is at the higher potential?
Am. 3.6 X I0* volts; toward right: 1',. > l'„
5. A Ferris wheel is 100 ft in diameter. Ms axis of rotation i on a
northsouth line, (a) If the horizontal component of the earth's field is
2.00 X 10 _i weber/m* and the wheel is rotating at 2.00 rev/rain, what
i* the potential difference existing between the axle and the end of one
apoke? (b) Which i^ at the higher potential? Ans. 488 pv
(}. Assume thai this mom has a uniform magnetic field directed
vertically downward and of flux density B = 1.0 X 10" weber/m*.
(a) Determine the magnitude and direction of the force on an electron
138 Looking In: Atomic and Nuclear Physics
thai enters the room moving due east and at an angle of 30° above the
horizontal with a speed of 3.0 X 10* m/sec. What, will be the speed of
this electron 1.0 X 10 '■' see after if enters the room? (b) An eastwest
wire is stretched horizontally across the room. What will be the direction
and magnitude of the force on :i (i.Om section oT the wire when there is
a current, of 12 amp westward in the wire?
Am. 2.4 X 10" newton, south; 3.0 X 10* m/sec; 7.2 X 10" 1 newton,
sou til
The most important thing a young man can learn from his first course in
physics is an appreciation of the need for precise ideas. W. S. Franklin
A new principle is an inexhaustible source of new views.
Marquis de Vauvenargues
To succeed in science it is necessary to receive the tradition of those who
have gone before us. In science, more perhaps than in any other study, the
dead and the living are one. Charles Singer
11
The Electron
The electron has conquered Physics,
and many worship the new idol rather
blindly, H. Poincare
It is the purpose of this chapter to discuss the experimental evi
dence of the existence of the electron, some of the measurements
which have been made on it, and the limitations of the classical
freeelectron theory of the conduction of electricity in metals.
11.1 IDENTIFICATION OF THE ELECTRON
In a paper "On Cathode Rays" (1897), J. J. Thomson first estab
lished the existence of free electrons. Thomson's investigation was
prompted by ihe divergent opinions people had at that time about
the nature of cathode rays. Experimenters had shown that at
low pressures (about 0.01 mm of mercury) air becomes a good
conductor of electricity and that, the discharge of electricity
through a gas produces light whose color depends on the gas and
in a pattern which depends on voltage and gas pressure (Kig. 11.1).
Some people considered the rays charged particles; others
thought of the display as a phenomenon in the "aether." Thomson
139
140 Looking In: Atomic and Nuclear Physics
suggested that an explanation based on particles as :i working
hypothesis was more likely to he successful and could he more
easily tested hy known laws (of mechanics) than any explanation
based on properties of the aether about which little was known.
Mr therefore devised experiments "to test some of the conse
Crooke's dork space
Cathode / _
Faraday dork spoce
7
# !■ %
Positive column
A
Anode
Negative glow II Anode glow
To pump
Cathode glow '
Fig, 11.1 Discharge of electricity through a gas at reduced pressure.
quences of the electrified particle theory." The objects of Thom
son's experiments were:
1. To verify that cathode rays carry a charge, a charge which
accompanies the rays when they are deflected by a magnetic
field
2. To investigate quantitatively the deflection of cathode rays
in an electric field, which deflection also indicates the presence
oT charge
:?. To determine the energy of the cathode rays and, by using
this value with data on the magnetic deflection, to deter
mine the speed and the ratio of charge to mass e/tn, for the
"particles"
4. To determine speed and <■ hi also from a combination of elec
tric and magnetic deflections
Thomson obtained information on each of these properties in
the following ways:
1. In a tube such as shown in Fig. 11.2, cathode rays leave
the cathode C, pass through an opening in the anode A, and reach
a region B where they are deflected by a magnetic field, pass
through an opening in a grounded cylinder 6, and finally reach a
collecting conductor /' mounted inside that cylinder. An increase
in charge is registered by an electrometer connected to /' only
when rays enter the opening in Q. The observations prove that a
charge is inseparably connected with the cathode rays and that
it is a negative charge.
The Electron
Mi
2. Deflection by an electric field was investigated in the tube
shown in Fig. 11.3, the precursor of our modern cathoderay
oscilloscope. It verified the negative charge of cathode rays.
3. The energy of the cathode rays was measured in a tube
without deflection plates and provided with a screened electrode,
Fig. 11,2 Discharge tube far demon
strating the (negative! charge and
magnetic deflection of cathode rays.
The magnetic field is excited in space
S by coils placed outside the tube.
as in Fig. 11.2. The innermost, electrode contained a thermo
couple. Its increase in temperature in a given time was measured,
and simultaneous measurement was made of the charge from the
rays received at the thermocouple.
If, in the time considered, A r particles strike the thermocouple
each bearing a charge r, the total charge is Q = A>. From the
ri.e in temperature the total energy is known
Ek  S\mt:~ (i i.)
Fig. 11.3 Thomson's tube for electricol deflection of cathode rays.
142 Looking In: Atomic and Nuclear Physics
where m is the mass and v the speed of one particle. Next the
radius r is measured for the path described by the particle in a
magnetic field. The centripetal force is equated to the magnetic
thrust
™ = cvB
T
(11.2)
For each of the two flat circular coils used to produce the mag
netic field, B is given by Eq. (10.29)
o _ Mo/
B  "27
(11.8)
where J is the current and mu is the permeability of free space,
4w 10" 7 weber/ampm. It follows from the preceding equations
thai
me* Nmv 1 2E k
v —
and
erli NerB QnJ
2E k
m tB Qho2P
(11.4)
(11.5)
With different gases (air, II 2 , and CO.) in the tube, Thomson
showed that e/m had the same value 2.2 X 10" coul/kg. Thus
cathoderay particles are independent of the nature of the gas.
4. Values of e/m can be obtained by a different method for
comparison with the foregoing results. In the tube of Fig. 11.3,
a magnetic field is established, into the plane of the paper, by
two coils whose diameters are etjual to the length of the capacitor
plates. The crossed electric and magnetic fields are adjusted to
give the cathoderay zero deflection. The cancellation of the
electric force (upward) and the magnetic force (downward) is
expressed by
isV — Bicv
(11.0)
where A'i is the electric field intensity.
Next the particles are deflected by a magnetic field only,
directed perpendicular to their velocity, and the radius r of the
path is determined from observed deflection on the screen. Here
— = Btfo
r
(11.7)
From Eqs. (11.0) and (1 1.7)
e v Ei
m
v
,B,
rBJi,
The Electron 143
(11.8)
With this method, and using air, C0 2 and H 2 , Thomson obtained
similar values for e/m. Again, the nature of the gas did not
influence e/m.
11.2 THE CATHODE RAY TUBE
We digress to point out the relationship between Thomson's
apparatus and the modern cathoderay tube. If the potential to
be observed is applied to plates !\ (Fig. 11.4) and a potential
o
p.
(a)
Fig. 11.4 (o) Cathode ray tube, (b) Test and sweep
potentials.
that increases linearly with time is applied to the horizontally
deflecting plates /■*., the electron beam (cathode ray) traces out
a wave form. If a sawtooth wave form is repeated at the same
frequency as the alternating potential on P t , the trace oi\ the
screen is repeated each cycle and appeal's to be stationary.
144 Looking In: Atomic and Nuclear Physics
11.3 ELECTROLYSIS AND THE ELECTRON
It has l>een seen (Chap. 8) that the combining properties of the
elements can l>o interpreted in terms of an atomic theory of
matter. Faraday's study of the electrolysis of aqueous solutions
of chemical compounds suggested thai electricity is also "atomic"
in nature. Faraday's discoveries of fundamental importance may
be expressed thus:
1. The mass of a substance separated in electrolysis is propor
tional to the quantity Q of the electricity that passes.
2. The mass ;!/ of a substance deposited is proportional to the
chemical equivalent k of the ion, that is, to the atomic mass A
of the ion divided by its valence t>,
MkQ^Q
(11.9)
where /"' is a constant of proportionality known as Faraday's
constant.
Careful measurements have been made of the amount of
electricity required to lilierate a mass of any substance numeri
cally equal to its chemical equivalent, say 107.88 kg of silver,
1 .008 kg of hydrogen, or 05.38/2 kg of ainc. This value is 9.052 X
10 7 coul/kmole. It is called 1 faraday. It is represented by F in
Eq. (11.9).
Faraday inferred from his experiments that the same definite
amount of electricity is associated in the process of electrolysis
with one atom of each of these substances. He considered that this
charge is carried by the atom, or in some eases by a group of
atoms, and he called the atom or group of atoms with its charge
an ion. In 1874, Stoney stated the hypothesis that "nature pre
sents us with a single definite quantity of electricity." He sug
gest ed the name electron for this quantity and calculated its value
from the faraday and from Avogadro's number N*. In terms of
values now accepted,
/.' = jy,«e . ij,cjf)2 x 10' coul/kmole
9.052 X 10' coul /kmole _ „ .
e ~ 0.0219 X m molecules/kmoie , ' 6021 X l ° C ° Ui
The Electron 145
11.4 CHARGE OF AN ELECTRON, MtLLIKAN'S EXPERIMENT
Millikan, about 1909, devised a highly precise experiment based
on the fact that electrically charged droplets of oil can be held
stationary between the horizontal plates of a capacitor by adjust
ing the voltage between the plates so that the weight of the drop
is balanced by the force due to the electric field. The "oil drop"
experiment can be used (1) to show that electric charge occurs in
multiples of a discrete amount and (2) to measure the value of the
smallest charge, the electron. See Fig. 1 1.5, page 147.
An oil drop will fall with accelerated motion until the drag due
to air viscosity becomes great enough to balance the weight of
the drop. For the small speeds which occur in this experiment,
the frictioual drag is proportional to the speed of the drop. Setting
the weight mg equal to the f fictional drag fct>) gives us an expres
sion for the terminal speed vi reached by the drop falling
mg = h'i (11.10)
To evaluate k, we take from hydrodynamics Stoke'slaw/.' = t'wnjr,
where ij is the viscosity of air and r is the radius of the drop,
assumed spherical.
If the oil drop has a charge c/, due to an excess (or deficiency)
of electrons, and if a uniform electric field is now established
between the capacitor plates, the electric force on the drop is
FftJ,
(li.ll)
When the drop now falls under the influence of gravitational and
electric fields, it attains a new terminal velocity v^:
F — mg = ft»j O'o up taken as positive) (11.12)
Combining Kqs. (11.11) and (11.12) gives
V
,<} — mg = hit (11.13)
Suppose now that, owing to random ionization, the charge on
the drop increases by an amount q„. There will be no significant
change in mass, but a new terminal speed v 3 will result from the
change in electrical force:
y
j (<1 + 9») — mg = Icvs
(11.14)
146 Looking In: Atomic and Nuclear Physics
From Eqs. (11.13) and (11.14), we have a measure of the change
in charge in terms of observed speeds:
Qn = y &(>»  *>j)
(11.15)
Millikan observed that experimental values for q„ were always
wholenumber multiples of a certain quantity. Pie inferred that
this quantity is the basic unit of charge, the electron.
Example. By timing its full through a known distance, an experimenter
determines "the successive prods y„ and c„ +1 of a single oil drop having
successive different random charges. He computes the change in charge
(q„) from Kq. (11.15) as tabulated below. What do these data indicate
for the charge of an electron?
By inspection, one notes that the values for </„ are, within experi
mental uncertainty, wholenumber multiples of 1.6 X 10"". By divid
ing this number into the charges q„, we find the values n in the second
column. The experimental value for the electronic charge is (hen the
average of the values of e in the last column.
q,„ X10"
coul
1.6 X 10~"
n
qn
e = — »
n
X 10 "coul
4.76
2.98
3
1.59
3.21
2.00
2
1.61
4.96
3.10
3
1.65
8.07
5.05
5
e
1.61
= 1.61
11.5 ELECTRONS IN METALS
The conduction of electricity through metals is fundamentally
different from electrolytic conduction. When a copper wire has
carried an electric current, even for a long time, no chemical
change can be detected in the copper. .More than a century ago
it was first assumed that electricity is an agent that can flow
freely in a metal. The sign of this charge and its direction of How
were unknown, but the flow was assumed to be from the arbi
trarily defined positive (highpotential) terminal to the negative
(lowpotential) terminal. At present we have evidence that elec
trons flowing in the opposite direction are responsible for the con
duction of electricity in metals. These electrons are called free
electrons because they are temporarily detached from atoms.
The number and freedom of motion of these electrons determine
The Electron 147
the properties of the material as a conductor of electricity (and
heat).
Xo other property of solids has such an enormous range of
values as does electrical conductivity (Fig. 11.6). The best con
ductors have electrical conductivities greater than that of the
best insulators by a factor of 10".
^=feo
v Telescope
Fig. 11.5 Millikan's apparatus for determining the charge of an electron.
Conduction electrons have been compared to a gas which is
free to move within the metal, under the influence of an applied
electric field. The metal is visualized as consisting of an assembly
of stationary positive ions permeated by an electron gas which
makes the metal as a whole neutral. This qualitatively attractive
picture proves to be inadequate in several important respects.
first, on this basis we should expect the specific heat of a metal
to consist of two parts: that of the ionized atoms considered as
vibrators (3/2) and that of the electron gas (}!£), or a total
specific heat of f ft. This finding is in marked disagreement with
10 _
10"
10°
io J
10 6
10'
to"
io ,:
Ohmcm
10" 10*
=f
Cu Ni Ho
Si
Ge
n i i i r
Se CU2O 8 Celluloid Mica
ZnO
Glass
Amber Paraffin
Porcelain
Quortz Ceraml
Semiconductors 
■ Insulators ■
■* Conductors™
Fig, 11,6 Resistivities (ohmcentimeters) vary over the enormous range of
aboyt lO. 2 "
Looking In: Atomic and Nuclear Physics
148
measured specific heat, of dements in the solid state, namely,
5S cal . P ( mole degree) = 3/f . a relation discern! by Du hmg
and Petit as early as 1810. Second, the variations of e tec tncal
conductivity with temperature and the escape of electrons
through a metal surface at high temperature (thermion.c emis
sion) cannot be explained quantitatively on a classical e c ct on
gas theory, neither can the great variation m conductivity.
Quantum theory (Chap. 16) provides a satisfactory way out of
these difficulties.
QUESTIONS AND PROBLEMS
1 (a) What Ls the kinetic enemy of at, electron which moves in a
oirce" of radius 5.2 em perpendicular to a magnetic mi uction of 9 .1
TlO^weWom*? (6) 523 potential difie^nce would be reqmred to
.top this electron after i, ^fctff'fi^ ,j X l0 ' volts
2 Vn electron of mas « and charge B falls through a potential
difference 1' and .hen enters at right angles, a region of uniform mag
fiS*. 8. What is the radius of the electron pa* m^mag
" L,t 3 Etetrom traveling 2.00 X 10' m/see are subjected to a magnetic
i,uhatb f 0.0030 weber/m» in the apparatus „^ by Thomson
£K3> to Snee/Ma) The capacitor plate '■^««J"J
wSt voltage imiM be applied to them to return the beam to its m
J ^teJ 1 StioBl (») SL in a .ketch the ^*£~«*»
eleclrnu velocity r. the maguel.c induction ft and the Uu tnUuld fc.^
4. An oil droplet of mass 2.5 X 10"" gm. which <^ "MH^d
2 electron charges, is between two horizontal capacitor p ate* 2 em
JpSZnS ft* *e droptef is entirely.snpported by etectne foraefe
from falling, what must be the potential ddTerenrc holi v^^Pjg
10 rft *SKb photograph, an ejeotron pato* bjrfWo
a circle of 12cm radius bv a magnetic induction of O.OOsO weber/m^
?«i<W»lutP the enemy of the electron (*) Calcdate tho enemy of an
Hec.ron whose path radius is 20 cm in the *.,,,, {«• d
/Ins. 3.6 X 10' 6 joule or 2.2 Mev, ^.U X iw J"""
6 V narrow beam of electrons traveling with speed e along toe
x & t pa s^s etwecn the horizontal plates of a parallelplate eapacrig
F ,3 Htamed lo a potential difference V. The separation of the
plans 111 Show that in traveling a distance ( in the capuclor the beam
will experience a deflection from J lie x axis, given by
„ t V I
9 =  T ~\
12
Ions
and Isotopes
The most important trends in indus
try today spring from an increasing
knowledge of the properties of atoms
and their component parts.
David Sarnoff
The discovery of isotopes was foreshadowed by studies in radio
activity (Chap. 17) about 1007. The possibility that two different
radioelements might lie identical chemically was inferred by the
failure to separate certain ones by any chemical means available.
Also Thomson's study of positive rays (HJl.Tj in iondeflection
experiments yielded two lines for neon (atomic masses 20 and 22) :
yet no dilTereuees were observed in the optical spectrum of the
Ne gas. Thomson recorded his suspicion that "the two gases,
all hough of different atomic weights, may be indistinguishable
in their chemical and .spectroscopic properties."
12.1 MASS SPECTROMETERS
Boon after World War I, Dempster, Aston, Bainhridge, and others
devised instruments for determining both the masses of isotopes
149
150 Looking In: Atomic and Nuclear Physics
and their relative abundance. Although there are many types of
mass spectrometers,* a brief consideration of their common ele
ments should clarify the operation of any type.
A moving particle might be characterized by its velocity v,
its momentum mv, or its kinetic energy \rns. We may consider
arrangements of electric and magnetic lie Ids designed to sort
charged particles according to these properties.
Two types of energy selector are suggested in Fig. 12.1. In
the first, ions from a source 8 are accelerated through a potential
i.ri
J""\
{a)
Fig. 12.1 Energy selectors.
(6) — :—
difference V. They acquire kinetic energy ^mu 2 equal to Vq. The
speed of an emerging ion is then
\ m
(12.1)
We have tacitly assumed that the ions are at rest at s. Actually
they may have small (I ev) energies of thermal motion. This is
usually negligible compared with the energy {say, >100cv) ob
'ained from the electric field. But for some purposes, the slight
spread in the velocity values for emerging particles might have,
to lx> considered. In Fig. 12.16, the ion beam passes between the
plates of a curved capacitor. The ions are acted upon by an elec
tric field /■' in the direction of O. The ions move in a circular arc.
The centripetal force is provided by the electric field:
mv* r.
 W = Eq
(12.2)
* in a mass spectrometer, a meter measures an ion current; in a mass
Spectrograph, the record is obtained on a photographic plate.
Ions and Isotopes 151
This equation can be rearranged to show that the device is an
energy selector
&nv* = %REq
(12.3)
In the selector of Fig. 1 2.2, the beam of positive ions passes
through a region where a magnetic induction B is directed out
Fig. 12.2
Momentum selector (B is directed out;
/V'.lJ
E = 0).
,' A. R
r < * v «
bid N o
ward from the page. By equating the centripetal force to the mag
netic side thrust, we get
mi
li
= qvB
(12.4)
and it is apparent that this tie vice is a momentum selector
mv = RBq (12,5)
Consider next a twain of positive ions acted on by an electric
field li and a magnetic induction B at right angles to each other.
In the situation of Fig. 12.3, the ions experience an upward force
B„„,
Fig. 12.3 Velocity selector.
1 +
— *■
I
fr.
I
qE due to the electric field. With B directed out from the page, 1 he
magnetic side thrust on the ions is qvB, downward. Only those
ions for which
qii = qvB
(I2.fi;
will pass through without any deviation. This filter selects ions
of a particular velocity
E
S = B
(12.7)
152 Looking In: Atomic and Nuclear Physics
I» a Baiubridge mass .spectrograph (Fig. 12.4), positive ions
from an ion source arc colli mated by slits s, and x* and then pass
into a region in which they experience an electric force to the left
and a magnetic force to the right (supplied by Hi which is directed
into the page). From Etj. (12.7), the only ions which pass through
slit « 3 are those of speed
r =  Bi
(12.8)
where /; has Iktu expressed as V ".,. Beyond s ;i the ion is influenced
only by a uniform induction B» (into the page) which causes the
Fig. 12.4 Boinbridge moss speetro
groph.
ion to move in a circular arc of radius ft until it strikes the photo
graphic plate /' where it makes a developable trace. By measuring
the distance 2/1 from this trace to slit s a , and using Eq. (12.4),
we can find the charge mass ratio
'/ _„ ''
m BJl
V
B y B#R
(12.9)
The charge on the ion will he a multiple of the charge on the,
electron (usually U or 2c), which the experimenter must find.
With q known, he can calculate the mass m of the ion. lie may
then add the mass of the missing elect ron(s) to (hid the mass of
the atom.
For measurements of highest precision (a few parts in I0 7 }, a
mass spectrometer is designed to cover only a limited region in
Ions and Isotopes 153
the mass scale from I to 2">0. Unknown masses are found by inter
polation between known masses (often using ions of molecules as
well as of atoms). Also, atomic masses can be deduced from the
energy release in certain kinds of nuclear reactions (Chap. 18).
A table of the "Ixst" values of atomic masses obtained as aver
ages of mass spectrometer and nuclear reaction data, adjusted
for selfconsistency, is useful in many calculations in modern
physics. *
12.2 ISOTOPES
As the accuracy in measurement of atomic masses increased, it
was established that not. all atoms of the same element have the
same mass. Atoms of the same element (same Z) which have dif
ferent masses are called isotopes. Many elements (Be, F, Xa, Al,
P, Co, etc.) occur naturally with only one isotope. Many others
(H, He, Li, B, etc.) have two, and tin, the most varied, has no
less than 10 isotopes.
It the mass of a carbon atom is taken to be exactly 12, then
the masses of the other elements, determined by quantitative
chemical analysis, come out to be nearly whole numbers. Histori
cally this led to Front's hypothesis that all elements were built
from hydrogen. This picture was spoiled by certain atomic masses
determined chemically: 35.8 for CI, 63.54 for Cu, etc. But when
measurement of isotope masses became possible, it was found
that the mass of every isotope of every element was very close in
an integer on the scale in which carbon is taken as 12 antra (or,
originally, oxygen defined as 10 amu). Naturally occurring chlo
rine, for example, is a mixture of about 7o per cent of an isotope
:S4.!)7!K) and 25 per cent of an isotope 38.9773. Its average mass,
as found in chemical experiments, is then §(35) + t($~) = 35.5.
The wholenumber rule may be retained in this form: The
mass of every isotope of every element is well within 1 per cent of
a whole number when expressed in atomic mass units, defined by
taking carbon as 12 amu, exactly. We thus retain the picture of
all atoms built up of some unit of which there is 1 in hydrogen,
4 in helium, Hi in oxygen, etc. We have yet to explain, however,
* Sec (.'. II, Blunthard, C. R. Burnett, H. G. Stoncr. and R. L Weber,
"Introduction to Modern Physios," appendix fi, pp. 392 100, l'rciitire
Hall, liic, Engleivood Cliffs, X. J. , 1958.
154 Looking In: Atomic and Nuclear Physics
why the atomic masses arc not exactly integers instead of being
very nearly integers.
QUESTIONS AND PROBLEMS
1. The values of E and B in the velocity selector of a mass spec
trometer are MIDI) colts, n. and 0,050 weber/m*. (fl) Wluit will be the
speed of ions passing through this selector? (b) By what radius will a
singly riiarsnl ion of mass 50 amu be deflected by a magnetic field of
2.5 X 10~ a weber/m 1 after leaving the velocity selector?
Arts. 1.6 X 10' m/aee;2.1 em
2. A singly charged positive ion is accelerated through a potential
difference of 1,000 volts, li is then subjected to a magnetic field of
0.10 weber/m 1 in which it is deflected into a circular path of radius
18.2 cm. (a) What is the mass of the ion? (h) What is the mass number
of the ion? Am. 2.05 X 10 ■'■ kg. 15.9S amu: If.
3. A dust particle has a mass of 3.0 X 10~ s kg and a charge of
5.0 X 10 10 coul. The particle is accelerated in an electric held until it
has a speed of 4.0 m/sec. (a) Calculate its kinetic energy in joules, (b)
What potential difference is required to give the particle this speed?
(c) If the particle moves at right angles to a magnetic induction of
0.20 weber/m 2 , what force will the particle experience? (</) What is the
radius of the circular path in which the particle will move in this mag
netic field?
Ans. 2.4 X 10"' joule; 480 volts; 4.0 X lO" 10 newtnn; 1.2 X 10 a m
4. In a Dempster mass spectrometer, positive ions formed by
heating a salt of an element are accelerated to a slit s, by a potential
difference V (about 1,000 vol Is). A narrow bundle of ions then passes
Photo, plote or
electronic detector
Fig. 12.5
through the slit into a semicircular chamber where there is a magnetic
induction II perpendicular to the ion velocity i> (Fig. 12.5). Ions having
different values of e,'m will travel arcs or different radius. Show that the
charge/mass quotient of an ion can be computed from
e
m
2F
fi ! r 2
Ions and Isotopes 155
5. On the photographic plate of a ma>s spectrograph, a trace made
by a singly charged ion is h.und jusf halfway between the line formed
by 16 (+) and that formed by CH,(+). Find (he mass of this ion if the
mass sped rennet er is (u) a Bainbridge type, when 1 the mass of an ion is
proportional In the radius m = kr, and (b) a Dempster type, where the
dispersion equation is »i = kr 1 .
Ans. 10.018148 amu; 1B.009074 amu
6. In a method devised by S. A. Goudsmit, masses of heavy ions
are determined by timing (heir period of circulation in a known mag
netic field. To get an idea of the timing requirements, calculate the
period of revolution of a singly charged ion of iodine ail 127 (mass
126.945 amu) in an induction of 0.045 weber/m 5 .
4ns, about 1.8 X I0~* sec
7. Show why the mass spectograph gives data on the atomic masses
of individual ions, while conventional chemical methods yield results
only on average atomic, masses.
8. Silicon has an atomic number of II. Consider two isotopes of
silicon having mass numbers 28 and 30, fill in ihe remaining spaces in
the table:
Mass number. , 28 30
Number of electrons in the atom
Number of positive charges in the nucleus
Number of protons in the nucleus
Number of neutrons in the nucleus.
13
Electromagnetic
Radiation
Electricity, carrier of light and power,
devourer of time and space, bearer of
human speech over land and sea, greatest
servant of man. Charles Eliot
I. CLASSICAL THEORY
In 1 8(14 , James Clerk Maxwell completed the structure of classi
cal electric and magnetic theory. His summarizing equations
stand with Newton's laws of motion and the laws of thermody
namics as masterpieces of intellectual achievement. The four dif
ferential equations show how electric and magnetic fields are
related to the charges and currents present and how  they are
related to each other. They correlated experiments in large areas
of physics and predicted important new results. Specifically,
Maxwell showed that a changing current will radiate electromag
netic waves in which E and li are perpendicular to each other and
to the direction of the wave motion. Tlis theory predicted that
electromagnetic waves of all frequencies should travel with tiie
speed of light c, whose numerical value can be determined by
156
Electromagnetic Radiation 157
measuring the force between currents. This theory was experi
mentally verified by Hertz in 1888, and by 1901 Marconi suc
ceeded in transmitting electromagnet ic signals across the Atlantic
Ocean. The. electric generator, motor, betatron, television, and
radar are based on principles included in Maxwell's equations.
Home of the relations and experimental facts which Maxwell
synthesized carry the names of earlier investigators,*
13.T GAUSS' LAW FOR AN ELECTRIC FIELD
Imagine a potatolike surface immersed in an electric field. The
flux *« of the electric field through this arbitrary surface is
measured by the number of lines of electric force that cut through
Fig. 13.1
surface.
Electric flux through o
the surface. Let the surface be divided into elementary squares
An small enough so they may be considered to be plane. An cle
ment of area can be represented by a vector As whose magnitude
is proportional to the area: the direction of .is is taken as the
outwarddrawn normal to the surface element.
The field intensity E is practically constant over an element
As; B is the angle between K and As. The flux is found by adding
up the scalar product A* ■ As cos 8 for all the elements into which
the surface has been divided:
*k = 2EAs (13.1)
For a more precise definition we replace the sum by an integral:
* K  /E • r/s (13.2)
The integration is to be taken over a closed surface.
* A reader not familiar with calculus notation may wish (o skhn
Sees. 13.1 to 13.4 and resume his study immediately after the statement,
of Maxwell's equations in Sec. 13.5.
158 Looking In: Atomic and Nuclear Physics
Gauss' law states that the net (outward) electric flux through
any closed surface is equal to l/*« times the net charge q enclosed
hy the surface :
eu $ K = q or «,,/£• ds = q (13.3)
Gauss' law provides a convenient way of calculating E if the
charge distribution is symmetrical enough so we can easily evalu
ate the integral in F.q. (13.3).
Example . A long copper tube of radius a has ;i charge of +q/l per
unit length. It is surrounded by a coaxial copper tube of radius !> which
earrics charge per unit length — q/l. Find the electric field (a) at a di*
tanee ri from the axis where 6 > r t > a and (l>) outside this coaxial
cable at, distance r : from the axis, Fig. 13.2.
Fig. 13.2
in) Draw a gaussian surface which is a cylinder of radius r u coaxial
with ihe cable. Since the electric field is radial, there will be no flux
through the cutis of the cylinder. For a length I of cable Eq. (13.3)
becomes
*u/E • da = t E(2irri)l  q
giving
(6) When the gaussian surface is a cylinder of radius ri, the net
charge within it is (+q/l  q/l) = 0, and hence E = outside the
outer conductor.
13.2 FARADAY'S LAW OF INDUCTION
A changing magnetic field produces an electric field, as described
in Faraday's law (Sec. 10.12). Consider a test charge (/ which
Electromagnetic Radiation 159
moves around the circle in Fig. 13.3, where a uniform magnetic
induction B is directed out of the paper and is increasing with
time. The work done on the charge, per revolution, is the product
of the emf S and the charge q. The work is also the product of the
Fig. 13.3 CKorged particle q moving in a
magnetic Induction 6.
force qE that acts on the charge and the distance 2irr. Equating
these two expressions gives
£ = 2iffli (13.4)
or, more generally,
8 = 6 E • dl (13.5)
If this last equation is combined with £ = —d$ B /dl, Faraday's
law of induction can be written
(13.G)
13.3 GAUSS' LAW FOR MAGNETIC FIELDS
Gauss' law for a magnetic field expresses the fact that in magne
tism there is no counterpart to the free charge q in electricity.
Isolated magnetic poles do not exist. Hence the magnetic flux
'l>; ( through any closed surface must be zero:
*b = <f> B • da =
13.4 AMPERE'S LAW
(13.7)
Ampere's law (Sec. 10.10) giving the relationship between current
i and magnetic induction li can be written in circuital form as
 <6 B ■ dl  i
Ho T
(13.8)
160 Looking In: Atomic and Nuclear Physics
The line integral can be applied to any closed path near the cur
rent; symmetry usually suggests the most convenient path.
Example. Find the magnetic induction Si al a distance r from a long
Straight wire currying current i.
Fig, 13.4 Magnetic induction S near a
long, straight wire.
Consider a circle of radios r centered af I lie wire for t he path of
integration. Since It is tangent i<> the circle al each point .4, vectors It
ami <l\ (the element of are) poinl in (lie same direction. From symmetry,
It has tiie same magnitude at each point on the wire. Equation (138)
becomes
 6(27rr) = ir
B = **
2irr
winch is the result obtained in Sec. 10.10.
Experiments show that just as a changing magnetic held
induces an electric field (Faraday's law, Sec. 10.12), a changing
electric field induces a magnetic field. Faraday's law for an in
duced emf G may be written
i
E
dt
(13.9)
The analogous expression for the magnetic induction produced
by a w hang ing electric field is
 <p II • (II = eg fjl
/"'■
(13.10)
where the constants n» and a, are required in the mks system of
units we are using. The situation expressed in Fq. (13.10) can be
visualized by considering the region between the plates of a
capacitor (Fig. 13.">a) which is being charged with a steady cur
rent i. The accompanying dl'I/dt produces a magnetic field: B is
shown For four arbitrary points in Fig. 13.56.
Electromagnetic Radiation 161
In considering the two ways of setting up a magnetic field,
(I) by a changing electric field dK dt and (2) by a current i, we
have assumed that there is uo current in the space considered in
(1) and that no changing electric fields are present in (2). But,
(a)
(b)
Fig, 13.5 The charging of a capacitor (a) produces a
changing field dE dt, which (b) produces a magnetic field 8.
in general, the contributions of both dE/dl and i must be con
sidered. Maxwell generalized Ampere's law, writing it in the form
££«.«
dt
(13.11)
The term t,\{d$y.'dl) has J he dimensions of current and is often
called the diaplawmvtit mrrint. Tims, although the conduction
current i is not continuous across the gap of a capacitor (because
no charge is transported across the gap), there is a displacement
current % D in the gap equal to the external i.
13.5 MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES
We assemble here the four basic equations we have just discussed :
*/ f =
0Erfl= ~ d
(1 3.3)
(13.7)
(13.9)
[  <£ n ■ di = t^%? + i (i3.li)
m<> / dt '
We have written .Maxwell's equations in the form they have when
I
162 Looking In: Atomic and Nuclear Physics
no dielectric or magnetic material is present, since we are chiefly
interested in radiation in "free space."
Consider that the start of an electromagnetic wave occurs at
the termination of a transmission line whicli is energized by an
oscillating electric circuit. Figure 13.fi suggests how the electric
field lines break away from an electric dipole as the charges
+q and —q first approach and then recede from each other in
successive time intervals of oneeighth of a period 7. An observer
at /' looking toward the antenna will "see" an instantaneous
electromagnetic field pattern with E down and B toward his right
t =
r/a
r/4
3T/3
Fig. 13.6 Radiation from an oscillating electric dipole. [Adapted from S. S.
A ft wood, "Efecfric and Magnetic Fields," John Wiley & Sans, Inc., New York,
1949.)
(into this page). A moment later, as the wave advances, these
directions will reverse.
If an observer is at a relatively large distance from the source,
the waves that move past hi in will be practically plane waves.
For the waves shown in Fig. 13.7,
B = B« sin (kx  wt) (13.12)
E = Eo sin (kx  at) (13.13)
Consider the rectangular prism I • I ■ dx to be fixed in space. Its
trace in the z.r plane is shown in Fig. 13.86. As the wave passes
over it, the magnetic flux <t?n through the rectangle will change,
inducing electric fields around the rectangle. The line integral of
Eq. (13.9) is dE h ; there is no contribution from the top or bottom
of the rectangular path where dl and E are perpendicular. The
Electromagnetic Radiation 163
magnetic flux *« for the rectangle is B(tdx). By differentiation
this gives
d$n , , dB
W = ldx dl
From Eq. (13.9) we have dE I = Idx (dB/dt), or
dE
dx
dB
dt
(13.14)
where we have changed to the d notation of partial derivatives to
indicate that both B and E are functions of x and ( but that in
n c y
Er<fE
N
i
B+rfB
dx
(c)
Fig. 13.7 to) Section of a plane electromagnetic wave troveltng to the right
E parallel to i axis, 6 parallel to y axis. The part marked dx is viewed in (b)
in the xz plane, and in (c) in the yx plane,
evaluating dE/dx, t is assumed constant (Fig. 13.7// is an instan
taneous "snapshot"). Also, x is assumed constant in evaluating
dB/dl at the particular strip in Fig. 13.7c. From Eqs. (13.12)
and (13.13),
kE cos (kx — oil) = o>Bn cos (kx — o>t)
and we have the relation
E _ CD _
F ~ & ~ c
where
angular frequency u> = 2ir/
wave number k = 2*/X
speed c = f\ = a>/k
(13.15)
164 Looking In: Atomic and Nuclear Physics
The ratio of the amplitudes of the electric and magnetic com
ponents is equal to the speed c of the electromagnetic wave.
In considering the trace of our rectangular prism in the yx
plane, Fig. 13.7r, we see that as the wave moves by, *« changes
with time and a magnetic field is induced at each point around
I d.r. This induced H is the magnetic component of the electro
magnetic wave; K and li each depend on the time rate of change
of the other. Since there is no conduction current £ in the space
considered, the line integral of Bq. (13.10) hecomes —iilli. The
flux *s through the rectangular element Idx is E{ld.t). By dif
ferentiation this gives
et*a , , dE
— tt = l ax jt
M dt
From Eq. (13.10)
an
dE
aJ"** ,0 "«
(13.16)
where partial derivatives have been indicated for the same reasons
as in Eq. (13.14). From Eqs. (13.12) and (13.13)
hllu COS {kx  ut) = — enflqfi'o COS (kx — <>)t)
and since c = w/k, we have
U = _L = l (13.17)
liu e<ijuiiw tviinC
By eliminating the fields between Eqs. (13,15) and (13.17), we get
c  — j— (13.18)
Substitution of numerical values in this relation gives
I
c =
v/(8.!) X 10 l2 eoulViitm ! )(4jrl0 ' weber, ,'ntm)
 3.0 X 10" m/sec
which is the speed of light in free space.
Maxwell made this calculation before it was recognized that
light was electromagnetic in nature and before Herts had detected
electromagnetic (radio) waves.
Electromagnetic Radiation 165
The magnetic part of an electromagnetic wave is often de
scribed by the magnetic field strength //, rather than by the mag
netic induction B. For waves propagated in free space, the only
case we shall consider,
H = B
(13.19
Since ju« = 4r X 10 7 weber/ampm. // has the dimensions
(weber m''i (ampm/ weber), or amp/ in. The quantity EH =
vVu/tu = 376.7 ohms is sometimes called the resistance of free
space.
13.6 ENERGY FLOW: INTENSITY OF A WAVE
Consideration of a parallelplate capacitor can lead us to an
expression of genera! validity for the* energy stored in an electric
field. To charge the capacitor to a potential difference A V requires
work W equal to AC AV*, where C is the capacitance. This energy
is stored in the capacitor a.s potential energy. Between the plates
there is an electric field E = AV/s, where « is the plate separa
tion. The capacitance is given in terms of the area A of the plates
and the permittivity e u as C  t„A/s. The stored energy can then
be expressed as
W = £C(AF) 2 = i*, EV = U E*As
Since .4s is the volume of the space between the plates of the
capacitor, we may associate the energy with the field, defining
the energy density of the electric field as the energy per unit
volume:
u, = (k) (u E* (13.20)
Similarly, consideration of the work required to establish a cur
rent in an inductive circuit suggests that the energy density for
a magnetic field is
„„.  ;„„//*  B* (13.21)
In a region where both electric and magnetic fields exist, the
energy density is given by
w = U lt E + ^H* (13.22)
166 Looking In: Atomic and Nuclear Physics
The rale at which energy is transported by a wave across unit
area per unit time is called the intensity .S' of the wave. Since the
wave speed is c,
S  c(i«P + innH*)
(13.23)
By the use of Eqs. (13.17) to (13.19), the energy density can be
written
w = s/tofie EH = ~ EH
c
and the equation for the wave intensity becomes
S = c EH = EH
c
(I3.1M)
(13.25)
The directions of E, H, and the velocity of the wave are properly
related if we define the intensity S as the vector cross product
S= ExH
(13.26)
This vector is called Poynting's vector. In mks units, it gives the
intensity in watts per square meter. Although we have derived
this expression for the special case of a plane wave, it is a general
relation which can be derived from Maxwell's equations.
Example. Consider a plane monochromatic planepolarized electro
magnetic wave (raveling horizontally northward, polarized vertically
(eleeirie field intensity directed alternately up and down). The frc
y (up) i 1
x (north)
Fig. 13.8
Z (east)
quency is 5.0 megacycles/see. The amplitude (maximum value) of the
electric field intensity E is 0/160 volt/m. (a) Give an analytical ex
pression for this wave, (6) Find the average intensity of this wave.
Electromagnetic Radiation 167
The wavelength is X = c/f = 60 m. The period T = 1// = 2jt/«
= 2.0 X 10 7 sec. If coordinates are chosen as in Fig. 13.8, from Eq.
(13.12) the electric field is
B, = B. 
E v = E B sin 2jr
(H)"
060 sin (irl0 7 f  0.105s)
The amplitude //„ of the magnetic field is
//„ = . p Bo = 0.060 volt/m — — ,  = 1.59 X 10 < amp/m
Vj«o 37 G./ ohm
Since the magnet ie field is confined to the eastwest direction,
//, =0 //„ = //. = 1.59 X 10* sin (jtIO'J  0.105s)
The Poynting vector S = E X H is in the x (north) direction, and its
magnitude is
5 = #o//u sin s 2v
(**)■
54 X 10«sin= (xlO'  O.lOox)
The average value of sin o over a cycle is .V, so the average intensity
of this wave is
S = hEoIh = 4.74 X 10» \vatt/m»
Maxwell's electromagnetic theory explained the thenknown
properties of light ; the experimentally measured speed of light in
free space, polarization, interference, and diffraction, and the
dispersion that occurs when light, passes through a medium where
wave speed depends on wavelength. Extended to x rays, the
theory identified them as also electromagnetic radiation, the dif
fraction of x rays by a crystal lattice being similar to the diffrac
tion of light by a ruled grating. Many seemingly diverse radia
tions were shown to be related regions of an electromagnetic
spectrum of grand extent — some 80 "octaves," of which the
visible spectrum comprises a little less than one octave.
II. QUANTUM EFFECTS
Beginning in 1900, developments took place which indicated that
Maxwell's theory does not predict accurately all aspects of elec
tromagnetic radiation and absorption of energy. These develop
ments led to the quantum theory. We shall trace the quantum
168 Looking In: Atomic and Nuclear Physics
hypothesis from its origin in explaining blackbody radiation, its
confirming success in explaining the photoelectric effect and
Compton effect, and its striking hut limited success in the Bohr
model of the atom (Chap. l;">) to its merging with other hypothe
ses in wave mechanics (Chap. 16).
13.7 BLACKSODY RADIATION
Any object continually emits and absorbs radiation, exchanging
energy with its surroundings. If the temperature of the object is
high enough, the radiation may be seen the material glows.
There is a direct relation between absorption and emission.
Kirchhoff's law stales that an object which absorbs radiation of a
Fig. 13,9
radiators.
Good absorbers are good
particular wavelength strongly is also a strong radiator at that
wavelength. Consider a platinum disk in a furnace (Kg. 13.9) at
thermal equilibrium. Suppose that it receives one unit of energy
per unit area per unit time and that the fraction p is reflected.
Then I — p is the fraction absorbed, designated a. Hut if the tem
perature of the disk is to remain constant, the disk must lose
as much energy per second as it receives. The rate of emission t
from the area considered must equal a. If a carbon disk is in
thermal equilibrium in the same furnace, it receives the same
energy per unit area per unit time as does the platinum, but it
absorbs a larger fraction (1  p') and hence must emit more,
to keep T constant. Good absorbers of radiation are good emit
ters, as represented by the relative lengths of the arrows for
p, p', t, and «' in Fig. 13,9.
Electromagnetic Radiation 169
All materials exhibit characteristic differences in their absorp
tion of radiation in different parts of the spectrum. (The colors of
many things we view are due to such selective absorption.)
Hence the emission spectrum for thermal radiation at a given
temperature depends on the material of the emitter.
We can imagine an ideal body which absorbs all radiation
incident upon it. By KirehholV's law, this body would also be
the most effective emitter of thermal radiation at all wavelengths.
Such an ideal absorberemitter is culled a blackbody.
Fortunately it is possible to realize blackbody conditions
experimentally to any degree of approximation requited. If we
form our material to make a cavity with a small opening to the
outside, the hole will behave as a blackbody. Radiation which
enters the hole will bounce around at the inner walls of the cavity,
Fig. 13,10 Cavity radiation Is nearly
blackbody radiation.
gradually being absorbed. Only a tiny fraction of the radiant
energy will be reflected back through the hole. Viewed from the
outside, the hole is an excellent approximation to a blackbody.
When the walls of a blackbody cavity are maintained at some
temperature T, the interior is filled with radiation. A tiny fraction
leaks out of the hole. A ray leaving the hole has, in general,
undergone many reflections. At each, reflected energy is added to
emitted energy until in the emerging (blackbody) radiation, the
energy distribution depends only on the temperature of the cavity
and not on the material of which its walls are made.
The blackbody radiation can lie dispersed by a grating, and a
bolometer or thermopile can be used to measure the energy radi
ated in each wavelength interval. A continuous spectrum is
found; that is, radiation at all frequencies is observed. The dis
tribution of intensity per unit wavelength interval is shown in
Fig. 13.11. When the temperature is increased, more energy is
radiated at every frequency, and the relative increase is greater
at the higher frequencies, shifting the maximum of the intensity
170 Looking In: Atomic and Nuclear Physics
distribution to lower wavelengths (Fig. 13.11a) or higher fre
quencies (Fig, 13, lib).
The area under the blackbody distribution curve f " S(f) df,
whirl] represents the total power radiated, is found i" increase as
the fourth power of the absolute temperature:
PeAT* (13.27)
where P is the radiant flux from a blackbody of surface area A
at absolute temperature T. This is known as the StefauBoltz
mann law ; the constant a has the value 5.7 X 10" 8 watt/ (m*) (°K*).
Intensity
I 3000 : K
Intensity
2.0 4.0
(o) Wavelength, microns
(6)
Frequency
Fig. 13.11 Blackbody radiation distribution: (a) intensity vs. wavelength, with
visible region dotted, (b) intensity vs. frequency, with Ti < T. < It.
13.8 PLANCK'S LAW
A number of physicists advanced theories based on classical
physics to explain the distribution of energy in the continuous
spectrum from a blackbody. Lord Raylcigh and Sir James Jeans
assumed that radiation in a cavity has degrees of freedom which
correspond to the frequencies of standing waves that are possible
in the cavity and that the energy is divided equally among these
different degrees of freedom. The resulting distribution law is
(RayleighJeans)
(1.3.28)
where K» is the radiancy (power per unit area) at a wavelength
X and ci and c a are empirical constants. The RayleighJeans law
fits the experimental data (Fig. 13.12) only for large values of
Electromagnetic Radiation 171
XT', fn fact, it leads to an "ultraviolet catastrophe" by predicting
that as X becomes smaller, R\ increases without limit; the total
power radiated by any body is infinitely large!
Wilhelm Wien assumed that cavity radiation came from
molecular oscillators among which energy was distributed with
respect to frequency according to a Maxwell distribution (similar
to the distribution law successfully used for molecular speeds in
the kinetic theory of gases). The resulting distribution law agrees
with experimental data in the shortwavelength region of the
Intensity
Planck's law
12 3 4
\Wgvelength, microns
Fig. 13.12 Agreement of radiation formulas with
Coblentz's experimental data (circles).
spectrum, but predicts values of I{\ which arc too low in the region
where X7' is large:
th = cVV— '" (Wien)
(13.29)
Max l'lanck also started with the assumption that the wallt
of a cavity radiator are made up of tiny electromagnetic oscilla
tors or resonators of molecular dimensions. He, too, used a Max
well distribution, taking e~ HlkT as the probability that an oscil
lator has energy /■.'. lie accepted the RayleighJeans calculation
for the number of oscillators per unit volume in the frequency
range from / to / + df. But Planck was led to make two radical
assumptions:
172 Looking In: Atomic and Nuclear Physics
1. An oscillator can have only energies given by
E — nhv
(13.30)
where c = frequency
k = Planck's constant (of "action")
n = an integer (now called a quantum number)
The equation asserts that the. energy of the oscillator is
"quantized."
2. An oscillator does nut radiate continuously (as expected on
.Maxwell's theory) but only in quanta of energy, emitted when
an oscillator changes from one to a lower of its quantized energy
states. The quantum (or photon) radiated has energy propor
tional to the frequency of the wave: •
K = hv
(13.31)
From these assumptions, Planck derived the distribution law
R*
2irAc 2
X 6 e*"* T  1
_CiX«
,,r, \r _
 (I'lanck)
(13.32)
where c = speed of light
k = Boltzmaiui constant
h = (Mi2"» X 10" w joule/sec
Planck's law has been written in the second form with empirical
constants Ci and c* for comparison with Eqs. (13.29) and (13.30).
For a wide range of temperatures (300 to 2000° K) and a wide
range of wavelengths (O.o to n2 ft), Planck's law represents the
experimental data within 1 per cent. It is interesting to note that
despite its initial success the quantum hypothesis was resisted by
Planck himself. Conservative in nature, he tried for years to
reconcile his "quantum of action" (A) with classical theory.
13.9 PHOTOELECTRICITY
bight or other electromagnetic radiation falling on the surface of
a metal (Fig. 13.13) can under certain circumstances liberate
electrons from the metal. The number of electrons emitted pet
second can he determined hy measuring the photoelectric current.
The energy distribution of the electrons can he determined by
applying a retarding potential and increasing it gradually until
Electromagnetic Radiation 173
the stopping potential V, is found for which no electrons reach
the collector.
The chief features oT photoemissiou are : (1 ) There is no detect
able time la<i (> 10 ' sec'i between irradiation of an emitter and
ejection of photoelectrons; (2) the number of electrons ejected
per second is proportions! I to the intensity of radiation, at a given
frequency; (3) the photoelectrons have energies ranging from
zero up to a definite maximum, which is proportional to the fre
quency of the radiation and independent of its intensity; (4) for
each material there is a threshold frequency v K below which no
photoelectrons are ejected.
* Light source
Fig. 13.13 Apparatus for photoelectric effect.
These characteristics of photoemissiou cannot be explained
by Maxwell's theory of electromagnetic radiation. In MM)">
Kinslein made the assumption that light of frequency c can give
energy to the elections in f he metal only in quanta of energy hv.
Fit her an electron absorbs one of these quanta, or it docs not.
If it is given energy hv, an electron may use an amount of energy
v> in escaping from the metal, where it has negative potential
energy, into the vacuum, where it has zero potential energy. The
quantity w is called the work function of the surface. The maxi
mum kinetic energy which the electron can have when it leaves
the surface is therefore
/■:
*. NIEIJS
= hv — w
(13.33)
This is called Ivinstein's photoelectric equation. It explains the
linear relationship E i: = an + h shown in Fig. 13.11: The slope a
measured from the graph agrees with the value of Planck's con
stant A; the negative intercept b is identified with the work func
174 Looking In: Atomic and Nuclear Physics
lion id of the metal. The intercept on the frequency axis is the
minimum frequency of light that will liberate electrons from the
particular metal. At this threshold frequency vt, the photon
delivers just enough energy to enable the electron to get out of
the metal (with E k = 0):
hvo = w
(13.34)
From Eq. (13.33), #*.„„,* is independent of the intensity of illumi
nation, in agreement with experiment.
The term photoelectricity includes several distinct phenomena.
In the external photoelectric effect (photoemission), electrons
*■!, M.
Fig. 13.14 Dependence of maximum energy of
photoelectrom on frequency.
are ejected from a solid (or liquid) surface into a surrounding
vacuum. Photomultiplier tubes use this effect.
Electrons and ions may he produced in a gas by ptiotoionista
tion. The ionization chambers used to detect x rays utilize this
effect.
Conduction electrons and positive "holes" which remain
inside a solid may be responsible for either photoconduction or
a photovoltaic effect. Photoconduction is a decrease in resistivity
under the influence of radiation. It is used in television camera
tubes and in control devices where an external battery furnishes
the electric power. The photovoltaic cell is a device for converting
radiation into electrical power. Kadiation acting on two dissimilar
layers in the cell gives rise to an emf in much the same way a
voltage is produced when Cu and Zn plates are dipped in acid in a
Electromagnetic Radiation 175
voltaic cell. Photovoltaic cells are used in photographic exposure
meters and in solar batteries.
There is also an internal photoelectric process, within an
atom, called the Auger effect, or autoionization. An xray quan
tum may be absorbed within the same atom from which it origi
nates, with the ejection of one of its electrons. The net effect is
thai the atom adjusts from an excited level to a lowerenergy
level, with the emission of an electron. Finally, there is an inverse
photoelectric effect in which an electron is absorbed by a solid
and a photon emerges.
The photoelectric effect gives strong support to Planck's
hypothesis that light of frequency v can be emitted or absorbed
only in packets of energy hv. The citation which accompanied the
award of the Xobel Prize to Einstein stated that it was for "his
attainments in mathematical physics and especially for his dis
covery of the law of the photoelectric effect."
13.10 THE CONTINUOUS X RAY SPECTRUM
X rays are electromagnetic waves of very short wavelength,
about 10"" to I0~" m. In an xray tube (Fig. 13.1")) a battery li
Fig, 13.15 An xroy tube.
heats a tungsten filament C so that it emits electrons. A potential
difference of several thousand volts between cathode (' and target
T accelerates the electrons. The fastmoving electrons are quickly
decelerated when they strike the metal target. Most of their
energy is converted into heat by collisions with atoms of the
target. But as the electrons are decelerated, they are expected to
176 Looking In: Atomic and Nuclear Physics
radiate, according to Maxwell's electromagnetic theory. The
Gorman term Itrrmxxtniltliniu is used for this "slowingdown
radiation." The radiation is emitted in all directions. When one
examines the beam of x rays emerging from the hole in a lead
shield, one finds a continuous distribution of frequencies up to a
certain maximum. This maximum frequency depends on the
potential difference at which the tube is operated: v,„„ x /AV = a
constant for a wide range of voltages.
The highfrequency limit in the continuous xray spectrum
is difficult to explain classically. It is easily clarified by the photon
hypothesis. An electron may suffer numerous decelerations as it
encounters various atoms in the target. ICach time a photon is
produced, whose energy hi> is equal to the decrease in kinetic en
ergy A A'* of the electron. Clearly, the highestfrequency photon
that can be produced is that which results from the complete.
conversion of the electron's kinetic energy into a single photon.
Since electrons arrive at the target with energy e Al".
hv mnx = e AV
(13.35)
From this Duane and Hunt law, Eq. (13.35), r'h may be deter
mined from the sharp cutoff of the xray intensity versus fre
quency curve at p„ k , x . There is good agreement with the ratio e/h
determined in other ways.
13.11 THE COMPTON EFFECT
Another even more direct confirmation of the photon hypothesis
came about <)23 in A. H. Compton's explanation of properties of
scattered x rays. Compton allowed a beam of monochromatic
x rays to fall on a hlock of scattering material such as carbon.
The scattered radiation was examined in an xray spectrometer
(an instrument which uses » crystal and an ionization chamber
to measure the wavelength of x rays incident on it J.
Spectrometer /\
C X
Fig. 13,16 Apparatus for observing
Compton scattering of x roys.
Electromagnetic Radiation 177
According to classical theory, scattered radiation should have
the same frequency as the incident radiation. Compton found
such unmodified radiation, but in addition he found a scattered
wavelength X' greater than that of the incident beam. The shift
In wavelength X'  X was found to increase as the angle 8 at
which the scattering was observed was increased (Fig. 13.17).
The scattering of x rays with increase in wavelength is called the
Fig. 13.17 Wavelength shift in
Compton scattering of x rays.
Wavelength, X — *"
178 Looking In: Atomic and Nuclear Physics
Campion, effect. The. plioton hypothesis provides a straightforward
explanation.
We describe the Compton scattering as an elastic collision of a
photon with a free electron which is at rest before the collision
(Fig. 13.18). We ascribe to the photon the "equivalent mass"
hvfe* (Chap. 14), and to this mass we attribute linear momentum
hv/c. The conservation of momentum may be stated in two
Photon
* Before After
Fig. 13.18 Compton scattering of a photon.
Electron
equations, since momentum is a vector quantity and the law of
conservation applies to each of the components:
kv hv „ ,
X component: — = — cos 8 + me cos #
c c
hv' .
y component: = — sin $ — mv sin 4>
(13.30)
(13.37)
where v' is the frequency of the scattered photon and me is the
momentum of the recoil electron.
Conservation of energy requires that
hv = hv' 4 A*
(13.38)
where K k is the final kinetic energy of the electron. Solution of
these three equations provides an expression for the wavelength
shift
X'  X =  (1  cos 9)
mr
(13.39)
which agrees with experimental data. The unmodified radiation
is interpreted as due to photons scattered by electrons strongly
bound in atoms.
Electromagnetic Radiation 179
Example. What is the change in wavelength <>r x rays Comii onscattered
in the backward direction (6 = 180°)?
X' X =
6.625 X 10 3t joule sec
9.1 X 10 3l kg (3.0 X IIP in sci")
= 0.048 X IO" 1 * m = 0.0484 A
(1  cos 180°)
13.12 WAVEPARTICLE DUALITY: PROBABILITY
We have discussed two theories of electromagnetic radiation.
The classical theory says iliat radiant energy flows continuously
as a wave. The wave theory gives a satisfying explanation of
interference, diffract ion. and polnrixa! ion experiments. The quan
tum theory says that radiant energy is exchanged in quanta of
amount Ac, whose value depends on the frequency c of the light.
This photon theory gives a satisfactory interpretation of many
experiments in atomic physics (blackbody radiation, photo
electric effect, the frequency limit in a continuous xray spectrum,
the Compton effect, and the line spectra characteristic of
elements).
In some ways these two theories are mutually contradictory.
The wave theory says that the photoelectric effect should show a
time lag when the light source has a very low intensity. The
photon theory when used to explain a singleslit diffraction pat
tern would have to assert that these particles arriving at certain
points on the screen would "cancel" each other. (How?) Although
each theory works well for its own experiments, something has
to "give" when we try to put the two theories together.
A resolution of this tlillieulty was suggested by a novel idea
proposed by Louis de Broglie, in his 1'h.D. thesis in 1924. From
consideration of relativity theory (Chap. 14), he deduced that all
particles must have a wave nature, just as light has a wave
nature. The intensity of the particle wave at any given point (or
the square of the wave amplitude) is interpreted as proportional
to the probability of finding the particle at that point. The
de Broglie relationship for the wavelength X of a matter wave is
X = *
V
(13.40)
where p is the momentum (mv) of the particle and h is Planck's
constant.
rr 1 *
180 Looking In: Atomic and Nuclear Physics
The exploration of waveparticle duality was continued in
M)28 by Max Horn. as follows. Kuergy is not distributed continu
ously throughout an electromagnetic wave: the energy is carried
by the photons. The intensity of the wave (which the classical
theory defines as energy flow) at a point in space is really a meas
ure of the probability of finding a photon there. The classical wave
has become a sort of guide for the individual quanta of energy.
We have resolved the waveparticle dilemma, but at the cost
of admitting that laws of chance govern the motion of micro
CDBEAEBDC
Fig. 13.19 Diffraction at a single slit.
Fig. 13.20 Doubleslit diffraction.
scopic particles. If we photograph a singleslit diffraction pattern
(Fig. 13,19), the relative intensities tell us that many photons
struck the plate in the region .4, a fair number in li, some in C t
etc. Very few hit near I) or /■.'. Suppose, now, we perforin the
experiment with one photon. We can predict that it has a high
probability of hitting near .1. a fair chance for li. less for ('. very
little chance of hitting near i) or K, Hut we 08JQ prediel to
which point the photon will actually go.
Consider an experiment in which a beam of electrons falls on
two slits (Fig. 13.20). The electron distribution at I 1 predicted
Electromagnetic Radiation 181
by classical theory is shown by the dashed line. The distribution
actually observed (solid line) is that predicted by considering
interference of the tie Hroglie waves. Now consider one electron
shot at a time. According to this wave picture, each electron is
represented by a single wave packet which divides equally be
tween the two slits. Vet if we place a particle detector at slii .1,
we never observe half an electron: we find cither a whole particle
or no particle. It is intriguing to try to devise an experiment that
would reveal the slit used by individual electrons, without de
stroying the interference pattern. No one has succeeded. If a
detector is placed at A, the interference pattern smooths out; the
classical result is obtained.
13.13 THE UNCERTAINTY PRINCIPLE
A consequence of quantum theory is that one cannot determine
simultaneously the exact position and velocity (or momentum)
Fig. 13.21 Supermicroscope.
o/vw
of any particle. As an example, assume that the exact momentum
of a particle is known. Then it has a definite wavelength X = h/p
and is a continuous plane wave of uniform "intensity." It is equally
probable to find the particle anywhere in space. At the other
extreme, assume that we have located the particle within a very
small region of space. Then its wave function is a short packet
that does not have any unique X. Hence the momentum is fuzzy.
The uncertainty principle predicts that in general we cannot
make a measurement on a system without disturbing it. for
example, suppose we try to "view" an electron with an (imagi
nary) supermicroscope .1/ (Fig. 13.21) to determine its position x
and momentum p. We may borrow an expression from optics
which says that the smallest displacement A.v the instrument can
182 Looking In: Atomic and Nuclear Physics
defect depends on the wavelength of the light and the halfangle
a subtended by the objective lens: Ax = X/(a sin a). We "view"
the elect ion by light which (liters the microscope anywhere within
angle 'la. This radiation, scattered by the electron, makes a con
tribution to the electron's momentum which is unknown by
Apz = p sin a — ^ sin a
If we write Ax as the uncertainly in position of the electron and
Ap* as the uncertainty in its momentum, combining the last two
equations gives
1 / X \ (k . \ h
Ax Ap x = I . — 1 [ : sin « J = 
a \sm a/ \\ / a
(13.41]
which shows that as we increase the precision of our measurement
of 3% the value of p becomes subject to greater uncertainty. (The
foregoing is offered merely to amplify the statement that wc can
not make a measurement on a system without disturbing it. The
numerical value of a depends on the criterion used for resolving
power.)
Werner Ileisenberg formulated the uncertainty principle in
1927 showing that from Sehrodinger's equation (Chap. Hi) /i/4ir
is the lower limit of the product of simultaneously measured
value of a particle's position and momentum:
Ax Ap > r
(\:i.m
Mere Ar and A/; are defined as vm deviations. There is an uncer
tainty relation only between certain pairs of variables, those
which are "canonically conjugate variables."* There are uncer
tainty relations, for instance, between position and momentum
(discussed above), angular momentum and angle, and energy
and time.
We have seen that quantum theory is significantly different
from classical theory in dealing with the interactions and struc
* See Condon and Odtshaw (wis.), "Handbook of Physics," cinq). I>,
McGrawHill Book Company, Inc., New York, 19oX.
Electromagnetic Radiation 183
tore of small particles, IT the quantum theory is correct, as we
think, there is no hope of understanding the elementary structure
of matter (atomic and nuclear physics) from the viewpoint of
classical physics. In the following chapters we shall use the ideas
of quantum theory. It will be interesting to see, however, that
there is a region between macroscopic and microscopic physics
where the laws of classical and quantum physics smoothly overlap
(correspondence principle).
The secret of education lies in respecting the pupil. R. W. Emerson
The most essential characteristic of scientific technic is that it proceeds from
experiment, not from tradition. Bertrand Russell
The most brilliant discoveries in theoretical physics are not discoveries of
new laws, but of terms in which the law can be discovered.
Michael Roberts and E. R, Thomas
Relativity Wonderland 185
14
tivistic mechanics. The insight it gives into the binding energy of
nuclei and the liberation of nuclear energy will he our cbier
interest. We shall look, also, at what relativity says about simul
taneity of events, time dilatation, and the aging of voyagers in
spaceships.
Relativity
Wonderland
The supreme task of the physicist is
to arrive at those universal elemen
tary laws from which the cosmos can
be built up by pure deduction. There is
no logical path to these laws; only
intuition, resting on sympathetic
understanding of experience, can
reach them. A. Einstein
14.1 NEWTONIAN RELATIVITY
In elementary experiments in mechanics we recognize that trans
lator? motion can be measured only as motion relative to other
material bodies, such as the workbench or the earth. Measure
ment of a speed involves measurement of both distance and time.
vt
S'
Fig. 1 4.1 Reference system S' moves with constant velocity v
in x direction relative to reference system S.
Relativity and quantum mechanics are two great theories of
twentiethcentury physics which have modified in remarkable
ways our ideas of the physical universe. For bodies traveling at
speeds close to the speed of light, Newtonian mechanics is replaced
by rclativistic mechanics. The relativity theory of the physical
meaning of space and time makes some simple predictions of
great importance. (I) The mass of a particle is shown to be vari
able, depending on the speed of the particle; (2) it is impossible
for any particle to have a speed greater than the speed of lighl ;
(3) mass and energy are interconvertible. In this chapter we shall
consider the evidence which leads to the formulation of rela
184
Up to now we have perhaps intuitively regarded time as a unique
variable, quite distinct from, say, space, energy, or the* behavior
of material things. \Ye might agree with Newton that "Absolute,
true, and mathematical time, of itself, and by its own nature.
Hows uniformly on, without regard to anything external." It
is helpful to formulate the kind of relativity implied by these
ideas for later comparison with the new relativity.
Consider a material reference body and some sort of timing
device (the rotating earth, or a crystal oscillator) to constitute
a spacetime system of reference for making measurements to
locate particles or to describe events. Now suppose a second
system of reference S' (Fig. 14.1) to be in uniform motion with
respect to the first reference system >S\ along the common line of
i heir x axes. Let the velocity of S' relative to S be v. Let us agree
to reckon time from the instant at which the two origins of coord i
186 Looking In: Atomic and Nuclear Physics
nates and 0' momentarily coincide. At any later time, the co
ordinates of 0' measured in system S will be x = vt, y = 0,
z = 0. An event which occurs at coordinates x,i/,z and time /
in system A' will, according to Xcwlon, have coordinates in system
S' given by
v*  y
X = z
f = t
(14.1)
14.2 THE AETHER
The wave properties of light were, demonstrated by Young,
Fresnel, and others during the first part of the nineteenth century
and were explained in Maxwell's brilliant theory of electromag
netic radiation (Chap. 13), It was difficult for scientists of the
nineteenth century, as for us, to conceive of a wave motion
without a material medium to transmit its vibrations. So they
invented a medium called the aether for the propagation of light.
The aether was thought to pervade all space, as well as trans
parent material bodies. The assumed existence of the aether sug
gested two interesting consequences worthy of experimental
check, (I) Light waves should travel with a definite speed
(c = 3.0 X 10" m see in "empty" space) with respect to the
aether itself. Then the apparent speed of light relative to a mate
rial body moving through the aether should be different from c
and should depend on the speed of the body. (2) An "absolute"
velocity of the earth or any other body should be ascertainable
from measurements on tight waves transmitted through the
aether.
14.3 MICHELSON'S INTERFEROMETER
An experiment designed to detect the motion of the earth relative
to the aether would require very sensitive apparatus, for the orbi
tal speed of I he earth is only about III ' the speed of the light
signals that would be used in the measurement. With this prob
lem in mind, Miehelson devised an interferometer, an instrument
in which interference patterns produced by two light beams are
used to reveal differences in the optical paths of the beams.
When the optical paths (Fig. 14.2) happen to he equal, beams 1
Relativity Wonderland 187
and 2 will arrive at E in phase and produce a bright field, by con
suuetive interference. As distance .1.1/ is increased X I by moving
mirror .1/, the optical path for ray 1 is lengthened by X/2, and
destructive interference of rays 1 and 2 at li gives a dark field.
r
c\
3w
~G\
M
Source
Fig, 14,2 Michelson's i nf erf ero meter.
Mirrors .1/ and .1/' are set nearly but not quite perpendicular to
produce a field crossed by alternate bright and dark interference
fringes (Fig. 14.8). These are counted as they move past a refer
ence mark R, For each fringe that passes the mark, the optical
path has changed by one wavelength. This change might he pro
duced by moving .1/ a half wavelength. However, the change
might also he produced by a change in the speed of light in beam
I (on the substitution, for example, of a gas of different index of
refraction for the air in that one beam).
Fig. 14.3 Fringes and a reference mark.
188 Looking In: Atomic and Nuclear Physics
14.4 THE MICHELSON MORLEY EXPERIMENT
Assume that the earth travels through stationary aether with a
speed i> and that light has a speed c in the aether. Consider a
Mieheison interferometer arranged so that one of its two equal
arms is parallel to the earth's velocity {Fig. 14.4). Then the. times
required for the light beams to travel the distances AM A and
AM' A will be unequal. The speed of a beam traveling from .1 to
.1/ is c — t" relative to the interferometer. On the return from .1/
M'
i
A
4—W£
i
M
n
HI
u
Fig. 14.4 Light poths in moving interferometer and velocity
vector diagram.
to A, the speed of the beam relative to the interferometer is
c + v. The time for the round trip AM A is thus
c — V C + V
2w
(14.2)
Since r is small compared with c, we may use the binomial
theorem to obtain the approximation
1 — rye 2 c \ c* /
(1U)
A wave front leaving A toward mirror .1/' will be returned,
according to Huygen's principle, but only after A has moved to
a new position .]'. The component of the velocity of light in the
direction perpendicular to the motion of the interferometer is
VV 1 — v i . The time for the round trip AM' A is
k «
2s
2»/
vV  v* Vi
!n /; _ 2* /
Iff*
2 ri
1+3^ +
)
(14,1)
Relativity Wonderland 189
Waves which are in phase when they reach A from the mono
chromatic source will differ in phase when they return to .4 after
reflection, because of the time difference:
Ai = ft — h =
si
(14. o
If the interferometer is rotated 90°, paths t and 2 will have
their roles interchanged and the total retardation will be 2sp !
The number of fringes passing the reference mark should be
N =
path difference _cA/ = c2st>* _ 2siP
wavelength X Xc* c 2 X
(14.6)
To estimate the magnitude of fringe shift to be expected, we
may assume that the earth's velocity through the aether is the
same as its orbital velocity, about HO km sec. By using multiple
reflections, .Mieheison and Morley attained an effective path s of
10 m (Fig. 14.5). For light of wavelength 5,000 A we should then
estimate a maximum fringe shift of
_ 2 X 10m(3 X lOWsec)' _ f . ( 4 ?)
N ~ OTX 10«m/»»)»(5.0 X 10 T m) U * lr " lge U4 ' J
A fringe shift of this amount is readily detectable with the
apparatus. It should then be possible to measure the fringe shift
and from it compute the velocity of the earth relative to the
aether, that is, the absolute velocity of the earth.
Surprisingly. Mieheison and Morley found no fringe shift
when the interferometer was rotated in a pool of mercury. It
appeared that optical experiments cannot detect motion of the
earth relative to the aether.
Mieheison and Morley reported their results in 1887. No
subsequent experimental evidence contradicts them. Some linger
ing doubts were laid to rest in a review article published in the
Reviews of Modern Physics (pages 107 178) in 1955.
Several attempted explanations for the apparent impossibility
of measuring the earth's absolute motion failed to gain acceptance
when they either did violence to established theory, disagreed
with known astronomical data, or introduced too many special
hypotheses.
m
Fig. 14,5 Michelson interferometer designed to detect "absolute motion" of
the earth, (o) Interferometer was mounted on a stone, ond floated in mercury
to damp vibrations and to permit rotation, (b) Multiple reflection of beams gave
an effective path length of 10 m.
190
Relativity Wonderland 191
14.5 POSTULATES OF THE SPECIAL THEORY
OF RELATIVITY
Consider several physicists in a completely enclosed elevator or
railroad car, moving with constant velocity relative to earth.
Could these people detect and measure the velocity of their
enclosure from observations made inside, with pendulum, spring
balance, etc?
Proceeding from considerations such as these, Henri Poincard
in the period 1 SiM) to 1904 developed the hypothesis that it is
impossible to determine absolute motions of a body or of a refer
ence system by any dynamical, electromagnetic, or optical means.
Measurement of the velocity of bodies relative to a stationary
net Iter seemed to Iks the best device classical physics could offer
for determination of "absolute" motion. The negative result of
the MichelsonMorley experiment was interpreted by Einstein
as indicating that only relative velocities can be measured. Con
sequently, the general laws of physics must be independent of the
velocity of the particular reference system of coordinates used to
state them, otherwise it would be possible to ascribe some abso
lute meaning to different velocities.
The special or restricted relativity theory of 1905 was limited
to consideration of reference systems moving at a constant
velocity with respect to each other {Fig. 14.1). Einstein based his
theory on two postulates:
1. The laws of physical phenomena are the same when stated in
terms of either of two reference systems moving at constant
velocity relative to each other {and can involve no reference
to motion through an aether).
2. The velocity of light in free space is the same for all observers
and is independent of the velocity of the light source relative
to the observer,
{The "general" theory of relativity, 1 9 Hi, is Einstein's theory
of gravitation and will not be considered here.)
Suppose person A. at rest in a laboratory, assigns to every
event which he observes a position (j.i/a) relative to a particular
origin fixed in his laboratory and a time J as indicated by a clock
at rest in his laboratory. Now let person B move through A's
laboratory with speed u in A's positive x direction. Let person
192 Looking In: Atomic and Nuclear Physics
B measure positions relative to an origin moving with him and
times with a clock (just like A's clock) also moving with him.
Then to each event B will assign a position (x',y',z') and a time I'.
Assume that the clocks are synchronized to read I = (' = when
the (x'y'z') axis momentarily coincides with the (xi/z) axis. The
relations which connect the distance and time intervals between
two events as measured from the two inertial reference frames are
x = —
V = y'
z = z'
x' + lit'
== X' =
U'  !J
Z' = 2
X — ut
» _ t_— (u/c*)x
Vi  1* 1 /**
(14.8)
These transformation equations were developed by Voigt
(1887) and Lorentz (1904) in exploring the aether hypothesis.
But Einstein showed that the transformations satisfied his rela
tivity hypothesis that the speed of light will be the same in each
coordinate system.
Example. Show that light has speed c in both the 8 and S' coordinate
systems.
Suppose that the light starts from x = 0, y = 0, t = at J = and
moves in the positive x direction. It will arrive ill the point x = A' at
the time X/e, time its speed through the laboratory is ,.. Person B will
observe the light to arrive at the point
x , m X  u(X/c)
Vl  u7c*
at the time
t , m X/e  (uM )X
Vl  m*/c*
The speed of light in the S' coordinate system is thus
x' X  (u/c)X
V = 7 =
I' X/e  (u/c*)X
= r
Relativity Wonderland 193
transport energy from one point to another with a speed exceeding
the speed of li<iht.
Several relations of particular interest will now be discussed to
illustrate the meaning of space and time variables.
14.6 VELOCITIES NEVER ADD TO MORE THAN c
Suppose that our two observers in coordinate systems S and S'
both observe an object which Hies past in the x direct ion. Observer
B measures the speed of the object relative to him as v' = dx'/dt'.
If we express v' in terms of the coordinates of the laboratory
observer A, we find
dt'
d[(x  mQ / V'1  u */c*] dx udl
d[[t  (*/c*)*]/Vl  »Vc*l di ~ (*/<&*
r^ws (l4! »
where in the last step numerator and denominator were divided
hy dt and dx/dt = » was written for the speed of the object in the
laboratory. Thus we have
»' + «
V =
V — u
or
v =
1 + uv'/c*
(14.10)
1 — uv/c 1
The speed v relative to the laboratory is not, as we might have
expected, exactly equal to the speed v' relative to B plus the speed
u of B relative to the laboratory.
Example. While observer B is moving through the laboratory with
speed u = 0.90c, a flying object passes him with a speed which he meas
ures as v' = 0.90c. What is the speed of the flying object relative to the
laboratory?
v =
V ' + «
0.90r + 0.90c
I.SOc
1 + w'/c* 1 + (0.90c) (0.90c)/c ! 1 + 0.81
= 0.994c
In other words, if a car were traveling at speed 0.90c, you would have
to drive at a speed of only 0.994c to pass it with a relative speed of 0.90r!
In a mathematical sense, the principle of relativity is that the
equations of physical phenomena must be invariant in form under
Lorentz transformations. The basic physical assumption of rela
tivity is that no mechanical or electromagnetic influence can
14.7 WHAT DOES "SIMULTANEOUS" MEAN?
Einstein pointed out by the following railroad story that man
cannot assume that his sense of "now" applies to all parts of the
194 Looking In: Atomic and Nuclear Physics
universe. He pictured a straight section of track with an observe?
seated on an embankment beside it. During a thunderstorm, two
lightning bolts strike the track simultaneously, at points Xi and
.)■•:. Kiustein asks: What do we mean by "simultaneously"?
x i A
Fig. 14.6 My time is not necessarily your time.
Assume that the observer is seated midway between r, and .<._>,
Assume that he has arranged mirrors so he can see x t and z» at
the same time without moving his eyes. Then If the reflections of
the lightning flashes are seen in the mirrors at precisely the same
instant, the flashes may be regarded as simultaneous, by ob
server A.
Now assume that a train speeds along the track and that
observer B on the train sits in an observation dome, with an
arrangement of mirrors for viewing points x x and x>. It happens
that observer B finds himself directly opposite A when lightning
strikes the rails at a and x,. Will the flashes appear simultaneous
to B? No, for if his train is moving from x, toward .c,, then the
flash at .10 will be reflected in his mirrors a fraction of a second
later than the flash in .r,. (In the limiting case with a train travel
ing at speed e, B would never see light from x«.) Whatever the
speed of the train, the observer B on it will always say that the
lightning Hash ahead of him has struck the track first.
In generalizing, we are forced to admit that two events which
occur at different places may be simultaneous for one observer
and not simultaneous for another. We cannot assume that a
single time scale (( = (') can be used with any and all coordinate
systems.
14.8 THE FITZGERALDLORENTZ CONTRACTION
To explain the null result of the MichelsonMorley experiment,
Fitzgerald in 1893 arbitrarily assumed a contraction of the arm of
the interferometer in the direction of motion of the apparatus.
Relativity Wonderland 195
The special theory of relativity predicts the same contraction
but ascribes it to the relative motion of the body and the observer.
Consider a material object in coordinate system S' whose
surface may be defined by the relation 4>{x' ,tf ,z') = 0. Then, by
the Lorentz transformation, the form of the. surface as viewed in
coordinate system S is
In particular, suppose that a spherical surface of radius a is
described in system S' by (Y) 1 + Cv') 2 + {*')" ' a" = <>■ Tbfe
appears in system S to be a moving ellipsoid
( x r*J + K
+ 31
whose semiaxes are (a \/\  u c,a.a). The surface undergoes
contraction in the direction of motion in the ratio y/\ — u s /c 2 : 1.
14.9 TIME DILATATION: THE CLOCK PARADOX
Consider now ihreflVel of relative motion on a flock. Two events
occur at a point in coordinate system .S": one at time t\, the other
at a later time 4 T <> an observer in S these events take place at
different points in space, (.ri,//,s) and Oj.(/,z), as well as at differ
ent times, such that {x»  xi) = u(t t  h). Prom the Lorentz
transformations
tt  h =
Thus the sequence in time of the two events is the same, but &t
appears longer for the observer in S than for the observer in .S".
This is interpreted as meaning that a moving clock appears to
run at a slower rate than does an identical clock at rest, in the
ratio Vl  «7c°:l.
The imminence of space travel has revived interest in the
"clock paradox" or "twin paradox." One of two identical twins
leaves his brother on earth and voyages at high speed into dis
tant space. On his return, he finds that his brother has grown
196 Looking In: Atomic and Nuclear Physics
much older than he, because of time dilatation in the spaceship.
Superficially, this is a paradox, for it challenges "common sense."
Also, it seems to contradict the assertion of special relativity that
in describing physical events all observers are equivalent; none
has a preferred or absolute reference system. The aging or clock
effect seems to provide a way of distinguishing among observers.
But, relativity asserts the equivalence of observers in inertial
systems, and since one of the twins accelerated at the start of his
space trip and again when he altered course to return, he did not
view his brother from the same inertial system before and after
the trip. So there is no paradox.
The intriguing question remains: Did the stayathome
brother grow older faster? Yes. In his 190") paper "On the Elec
trodynamics of Moving Bodies," Einstein wrote,
If at the points A and B there arc I ho stationary clocks which,
viewed by a stationary observer, arc synchronous, and if the clock at A
is moved with the velocity v along the line AB to B, then on its arrival
at B the (.wo clocks no longer synchronize, but the clock moved from
A to B lags behind the other which has remained at B by tv 3 /2c (up to
magnitudes of fourth and higher orders), ( being the time required for
the journey from A to B.
It is at once apparent that this result still holds if the clock moves
from A to B in any polygonal line, and also when the points A and B
coincide.
Bergman n has suggested the following elucidation of the clock
effect, tig. I 1.7.
9"^= — I"
Fig. 14,7 Clock paradox.
Observer A arranges for periodic light signals to go from
lamp L to mirror .1/ and back (a kind of optical clock). Light
travels a distance 2/) for each LM L circuit. Observer B is moving
with constant speed u at right angles to the line LM. For him,
Relativity Wonderland 197
the same light signal travels the larger distance 2D'. If observer
B set up a similar experiment in his coordinate system S', his
light signals would complete their round trips in shorter times
than noted by observer A. The discrepancies arise because the
two observers do not agree on which of two distant events {com
pletion of the nth round trip by either light signal) takes place
first.
Now let observer B suddenly reverse his velocity (u[). He
is now in a different Lorentz frame. (He accelerated.) His notions
of simultaneity have changed. Observer A sees B coming toward
him, with B's light signals arriving slower than his own. When
they meet, A's signals have completed a larger number of LM L
circuits than have B's signals. Observer A has aged more than B.
Example. What, will he the difference in I he rates of two identical
clocks, one of which is on a spaceship moving at 300 mi sec relative u>
the other?
u = 300 mi/sec = 5.25 X 10 s  m/scc c = 3.0 X 10 a m/sec
Relative change in rate
f. u* f, S^iTx 10" /  —  —
= 0.002 per cent, approx.
10"«
Experimental detection of time dilatation was achieved by
Ives and Stilwell (1938) on viewing the spectral lines of hydrogen
atoms which were given a high speed directed away from the spec
troscope. An arrangement was used to distinguish relativity
effects from Doppler effects. Light from the atoms fell on the
spectrograph slit directly, and also after reflection in a mirror set
at some distance and normal to the velocity of the atoms. Owing
to the Doppler effect, each spectrum line was split into two fre
quencies. Then light from hydrogen atoms at rest was viewed
with the same spectrograph. This gave lines slightly displaced,
in frequency, from the middle, of the Doppler pairs, in amount
predicted by relativity,
.Measurements of the lifetimes of mesons have been used to
cheek relativity predictions. The mean life of fi mesons (about
198 Looking In: Atomic and Nuclear Physics
'2 X 10"* sec) has been found to depend on I heir speed roughly
in (lie way predicted by relativity.
14.10 MASS AND ENERGY
Two results of relativity theory which are of especial importance
in atomic physics are (1) tlie variation of the mass of a particle
with its speed and (2) the equivalence of mass and energy.
Experiments have been performed, first by Bucherer in 1909,
on the deflection by a magnetic field of electrons whose speeds
are not small compared with the speed of light. The acceleration
may be determined from the radius of curvature of the path
a = v*/R. The force producing this acceleration is the magnetic
side thrust on the electron / = Bev. It is found that for high
speed electrons Newton's law in the form/ = ma is not satisfied.
But Newton's law written in the form / = d(mv)/dt is satisfied,
provided we assume that the mass m of the particle depends on
its speed. It is found necessary to assume that a particle which
has mass m u when at rest has a mass
m =
mo
y/\  i.'Ve*
(14.11)
when moving with speed v. The quantity wio is called the rest
mass. When v « c, m = mu.
Variation of mass with speed is accepted in relativity theory
as requisite for the conservation of momentum, which remains a
basic principle of mechanics. In order to have the total momen
tum of an isolated system remain constant, the momentum of a
panicle is delined as
p = mv =
Met)
Vl  »Ve*
(14.12)
Table 1 4. 1 shows for various ratios of v/c the kinetic energy of
an electron, the ratio vi/m , and the product BR, from which one
may get the radius R of the path in a magnetic induction B of
given value. Looking at the table, one might say that in problem
solving to sliderule accuracy, one can neglect relativity variation
of mass for bodies having speeds less than 0. 1 l he speed of light.
Relativity Wonderland 199
Table 14.1 Date on electrons
V c
Energy, ev
ffl fllfl
8R, X10 • weber/m
0.
1.00000
0,0100
25.54
1.00005
17.0
0.0200
102.2
1.00020
34.06
0.0500
638.5
1.00125
85.0
0.100
2,575
1 .00504
171.3
0.200
10,530
1 .02062
347.8
0.500
79,030
1.1547
983.6
0.600
127,700
1.25000
1,278
0.700
204,300
1.4002
1,669
0.800
340,500
1.6666
2,272
0.900
661,000
2.2941
3,517
0.990
3,110,000
7.0888
11,960
The kinetic energy of an object having speed r is equal to the
energy required to accelerate it from rest to the final speed v.
E h  //'(cos $) ds = jT dx (14.13)
But now we must use in place of/ — ma for Xewtou's second law,
d me
F
givmg
dt y/l  B */ c »
ft [* »g_ d X = m f r vd( ^=)
J dt y/\  r * J3 JO VVl  W/
This may be integrated by parts using the standard form
/« dv = uv — $v du
to obtain
ft = ffloiM — 7=  1 J
ft = »ic 2 — muc 2
(14.14)
(14.ir>)
This expression replaces the classical formula hn^ for kinetic
energy when v is comparable with c.
The equation for kinetic energy, Kk = (m — m.n}c' i , says that
when we speed up a particle, the increase in energy is propor
tional to the increase in mass of the particle:
&S = c(Am) (14.10)
200 Looking In: Atomic and Nuclear Physics
We can identify c times the relativist ie mass of the particle with
the total energy K of the particle:
E = mc * = A* + Wo c 2 (14.17)
Total energy = kinetic energy + rest energy
Kinstc.in's famous relation IC = mc % states that mass and
energy are different aspects of the same thing. It tells us tiie rate
at which one may he converted into the other.
Example. Find the energy equivalent of 1 gm of coal (or any other
substance).
E = me 2 = 0.001 kg (3.0 X 10* m/sec,} 1 = 9 X 10 13 joules
= 25,000 niogawuttlir
Only a liny fraction of this amount of energy is released in the burning
of I Km of coal: tin combustion products have a mass only sJightly less
than 1 Kin. In nuclear reactors, a somewhat larger percentage conversion
takes place, but it is still a small fraction.
14.11 NUCLEAR BINDING ENERGY
Mass spectrograph measurements show that the mass of any
stable isotope is less than the sum of the masses of its constituent
protons, neutrons, and electrons. Kinsfein's massenergy relation
suggests thai the mass discrepancy might account for the energy
needed to hold a nucleus together, against the dispersive forces
exerted by the protons on each other owing to their positive
charges.
The mass of the constituent particles for nucleus zX. A is the
sum of Z proton masses and (AZ) neutron masses. The mass
defect Ant is then calculated from
Am = Zm.fi + (A — Z)m„ — M z,A
(14.18)
where ma = 1.00814o amu, mass of the hydrogen atom
m„ = 1.00898b' amu, mass of the neutron
M x , = mass of the neutral atom of atomic number Z and
and atomic mass number .1
From A/i = (Awi)c 3 one can calculate that I amu is equivalent
to Oil I Mev (million electron volts).
Binding energy is primarily a property of the nucleus. Yet in
the equation above we have used data for neutral atoms >»u and
(14.19)
Relativity Wonderland 201
Mx..\ which of course include electrons. We justify this procedure
by the following facts: (1) If a nuclear reaction is written in terms
of the symbols for the corresponding atoms, the number of elec
trons on one side of the equation generally cancels the number of
electrons on the other side. (2) The minute changes in mass
which may accompany the formation of an atom from its ion and
electron(s) is negligible. (3) The mass data from mass spectro
graph experiments are always tabulated in terms of neutral atoms
(e.g., Na) even though deflection measurements must be made on
ions (e.g., Na ++ ). It is to avoid the trouble of specifying each time
the degree of ionization that the experimenter adds to his experi
mental value for the mass of the ion the proper number of electron
masses and reports as the isotope mass the* computed mass of the
neutral atom.
The binding energy per nucleoli is defined as the binding
energy divided by the number of nuclear particles:
Binding energy _ Am <■
Xucleon .1
It is this value which is significant in comparing the stability of
two different isotopes.
14.12 RELATIVITY AND SPACE TRAVEL
Rockets for space exploration require highly efficient sources of
thrust and large amounts of electric power. These requirements
suggest nuclear power sources. In this sense, the massenergy
.■elation of relativity is important to space travel. But other pre
dictions of relativity, such as time dilatation, are probably not
significant to space travel.
If we could burn nuclear fuel so efficiently that onetenth of
the initial mass of the spaceship were converted into kinetic
energy, the final speed would be less than 0.5c. This would give a
very small (0.14) time dilatation — hardly enough to allow one
generation of voyagers to reach destinations outside the solar
system.
PROBLEMS
1. An atom moving at a speed of 1.0 X 10" m sec ejects an electron
in the forward direction with a relative speed of 2.0 X 10" m/scc. Find
202 Looking In: Atomic and Nuclear Physics
the electron's speed as seen by an observer at rest (a) using a Newtonian
transformation and (6) using a I.orentz transformation.
Arts, (o) 3.0 X 10* m/sec, (6) 2.7 X 10 s m/sec
2, Find the length of a meter stick when it is moving at a speed
0.90c relative to the observer. Consider the cases when the stick is
oriented («) parallel and (6) perpendicular to its direction of motion.
Arts. («) 18.5 em, (b) 10(1 cm
3. What speed will an electron have to acquire for its relativity
mass to be twice its resi mass? Ans. 2.5 X 10 K m/sec
1. What is (he energy equivalent of the mass of an electron?
Ans. 0,51 Mot
5. What is the radius of curvature of the path of an electron whose
kinetic energy is 20 Mev when moving perpendicular to a magnetic
induction of 0. 10 weber/rn? Attn. 0.6S m
h. Imagine that you are moving with a speed c past a man who
picks up a watch and then sets it down. If you observe thai be held the
wmIcIi for (>,0 sec, how long does he think be held it? {Hint: You want
It — U when you know t' t — l\.) Ans. 4.0 sec
7. From the massenergy relation, calculate the energy released in
the reaction ,H= + ,H*  tHe*. (Data: ,H« = 2.014743 amu, Mr*
= 4.003874 amu) Ans. 24 Mev
8. A meson has a lifetime / = 1.0 X 10~*see before it decays. Find
how far a meson with t> = 0.09c can travel. Ans. 300 m
9. Find the energy liberated when an electron and a positron
annihilate. Ans. 1.02 Mev
10. If one uses the nonrehitivistic formula E,. = },m,r~. does one
overestimate nr underestimate the kinetic energy of a particle of rest
mass »io and speed vl
15
Hydrogen Atom
Bohr Model
... for the value of his study of the struc
ture of atoms and of the radiation emanat
ing from them. Nobel Prize citation
for Niels Bohr, 1922
By 1011, two rival pictures of the structure of an atom had
evolved. J. J. Thomson suggested a "currant pudding" model of
the atom iu which the positive charge was spread throughout a
spherical volume of radius about 10 ll> m, with electrons vibrating
about fixed points within this sphere. Ernest Rutherford sug
gested a nuclear model of the atom in which the positive charge
and almost the whole mass were concentrated in a very tiny
nucleus; the electrons roamed through the rest of the atom, out
to a radius of about 10~ 10 m. In crucial experiments, II. Geiger
and E. Marsden probed the atoms in thin metallic foils with fast
(" — sV c) a particles and showed that the observed deflections
could be explained by the intense electric field near the center of
a nuclear atom. Building on Rutherford's nuclear picture and
using Planck's quantum hypothesis, X. Bohr fashioned a model
of the hydrogen atom which explained its characteristic line
203
204 Looking In: Atomic and Nuclear Physics
spectrum and correlated this with electrical measurements of
excitation potentials and the ionization energy.
15.1 NUCLEAR ATOM REVEALED BY ALPHA SCATTERING
a particles are helium ions (He ++ ) and are emitted spontaneously
by some radioactive substances. In the Geiger and Marsden
apparatus (Fig. 15.1), a particles are directed against a thin
Fig. 15.1 Apparatus for investigating rt particle scattering, showing:
radioactive substance fi, the source of a particles, thin foil F of scattering
material, zinc sulfide screen S, and microscope M. a particles emerge from
a channel cut in the lead block I, strike foil F, and ore scattered to screen
S. The conical bearing allows rotation of microscope and screen about ver
tical axis FF. [H. Geiger and E. Marsden, The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science, 25. 604 (I913).l
metallic foil F in an evacuated chamber. The number of a par
ticles scattered at various angles with the original beam direction
is found to decrease with increasing angle, but some a particles
are scattered at angles greater than 90°, up to 180°. Rutherford
found this "almost as incredible as if you had fired a 15inch shell
at a piece of tissue paper and it came back and hit you." For the
IIe ++ ion is roughly 7,:i00 times the mass of an electron, and
therefore the large deflections of a particles cannot occur by single
collisions with electrons. Also, the foil used is so thin that a large
Hydrogen Atom— Bohr Model 205
a deflection cannot result from several successive collisions with
electrons. But, Rutherford reasoned, if all the positive charge
and most of the mass of an atom are concentrated in a very small
nucleus, then the a particle can come very close to a large amount
of charge all at once, and it will experience a large deflecting force.
Further, since the mass of the deflecting nucleus is greater than
that of the a particle, backscattering is possible.
Rutherford derived an equation for a scattering based on the
assumptions that the nucleus and a particle behave as point
Fig. 15.2
particles
atoms.
Deflection of a
by nuclearmade!
W^
Incident
a particles
Target
positive charges, that Coulomb's law applies to the mutual repul
sion even at small distances, and that ordinary Newtonian
mechanical principles hold (conservation of energy and conserva
tion of momentum). The number of a particles N reaching unit
area of screen at distance r from the scattering foil was predicted
to depend on
r 1 4VW AV »i» 4 <»/2)
(15.1)
where A' a = initial kinetic energy of a particle
N ( = number of a's incident per unit time on foil of thick
ness / having n target nuclei per unit volume
Z = nuclear charge
2e = a's charge
206 Looking In: Atomic and Nuclear Physics
In their precise and very readable report, Geigcr and Marsden
neatly tabulated the results of counting thousands of a particles
to show that .V was found to be proportional to (a) the thick
ness / of the scattering foil, (M the square of the nuclear charge
Ze (using foils of Au, Ag, Cu, etc.), (c) the reciprocal of sin 4
(0/2), where B is the angle of deflection, and ((f) the reciprocal
of the square of the initial energy K a of the a particles (using
different radioactive sources).
The Geiger and Marsden experiments verified Rutherford's
nuclear model of the structure of an atom. They clarified the
meaning of the atomic number Z and showed it to be more sig
nificant than atomic mass in ordering elements in relation to
chemical properties. An upper limit of 10 IS m was obtained for
the size of the nucleus, in terms of distance of closest approach of
a particles. The validity of Coulomb's law was verified down to
about this distance of separation between charges.
When Geigcr and Marsden used still more energetic a particles
in their deflection experiments, some deviations from the scatter
ing pattern predicted by Kq. (15.1) were observed. This was the
first hint of the existence of a "nuclear force" of attraction in
addition to gravitational force and the electrostatic (Coulomb)
force of repulsion.
15.2 DATA FROM SPECTROSCOPY
A grating spectrograph (Sec. 9.12) disperses the light incident on
its entrance slit and focuses on a photographic plate a line image
of the slit for each different wavelength present. As fine diffrac
tion gratings became available, owing largely to the skill of
H. A. Rowland (1848 1901), spectroscopists diligently accumu
lated a vast number of measurements on the radiation emitted by
atoms when excited in electrical discharge tubes, in ares, and
otherwise. In general they found that (I) each element has its
own characteristic line spectrum of wavelengths A or frequency v,
(2) spectrum lines are generally sharp; elements producing the
sharpest Hues are very stable; ('.',) lines in a sped rum may differ
in relative intensity and in degree of polarization; (4) to the
spectrum of every element can be ascribed a series of "term
values" such that the frequency of every observed spectrum line
can be obtained by differences of these term values.
Hydrogen Atom— Bohr Model 207
We shall consider the muchstudied spectrum of hydrogen,
the simplest atom. Its spectrum comprises several welldefined
groups of lines: the Lyman series in the ultraviolet, the Maimer
series in the visible region, the I'aschen series in the infrared, and
others still farther oul in the infrared, Fig. 15.3.
Balmer limit
Visible p asc hen limit
Lymon limit \ "■ * j
. '•« 1 1 i r i i .1
i_
j
4,000
8,000
1 2,000
16,000
20,000
Fig. 15.3 Some series of lines in the spectrum of hydrogen (wavelengths in
Angstrom units).
As a first step in developing au acceptable theory of atomic
spectra, Rydberg found a relation which allows one to calculate
the wavelengths in the hydrogen spectrum from differences
between terms:
l = R w ~ }y Khvve R = Lm x 10 ~ 3 A ~' < I5 ' 2 >
When rtf = I and «,■ is given successive values, 2, 3, 4, 5, . . . ,
the differences of the terms in the Rydberg equation give the
wave numbers 1 ,/X for the lines of the Lyman series. When n f = 2
and it; = :>. !, ."», .... the Rydberg equation gives the wave
numbers for lines of the Halmer series, etc. Although this formula
was obtained empirically, it turns out to be closely related with
the way the spectrum originates.
15.3 BOHR'S THEORY
There is a similarity of the hydrogen atom and our planetary
system, in that in each case there is an attractive force inversely
proportional to the square of the distance between the bodies.
Bohr accepted Rutherford's concept of the nuclear atom and
devised a model of the hydrogen atom in which orbital motion of
the election was used to predict wavelengths of radiation which
agree very closely with the observed wavelengths of the spec
trum lines (Table 15. 1),
208 Looking In: Atomic and Nuclear Physics
Table 15.1 Some term values end energy levels for hydrogen
Wovenos. (1/cm) Joules
*
Elec. volts
ty
H H
t>
O
■~
N
■nT
cm
r*>
•o
fo
CN
CM
O
<0
S
PI
in
Batmer series
u
ti
■j
u
^
00
4
i
CM
o
8
CM
00
&
d
3
s
3,047
4,387
6,855
12,184
27,419
6.0x10" 20 0.38
8.7
13.6
24.2
54.3
0.54
0.85
1.51
3.39
109,677
217.3
■13.58
man series
6562A
4861
4341
4102
Sol
tier series
15, 233c
20,264c
23,032c
24,373c
3646
(Limit) 27,419c
The following assumptions are made in the Bohr theory of the
hydrogen atom: (1) The electron moves around a stationary
nucleus (a good approximation, since m, iue = 1,830m,.). (2) The
electron is held in a stable circular orbit by the Coulomb attrac
tion between the negative electron and the positive nucleus.
(3) Only certain (quantised) orbits are possible for the electron,
namely, those for which its angular momentum is a wholenumber
multiple of h/2ir, where h is Planck's constant. (4) Radiation is
emitted (or absorbed) by the hydrogen atom only when the elec
tron undergoes an energy change in a transition from one orbit to
another. The energy of the photon emitted (or absorbed) is given
by
hv  K,  /:, (15.3)
Newton's laws of motion are assumed to lie applicable to the
hydrogen atom, just as to bodies of larger dimensions. The force
Hydrogen Atom Bohr Model 209
of attraction exerted by the nucleus on the electron has the
magnitude
F =
(l,,.w
(15.4)
The electron, moving with uniform circular motion, experiences a
central acceleration a e = r 2 /r and a centripetal force mr r, from
Newton's second law. We equate the Coulomb force and the
centripetal force
r
(4xe u )r 2
Hi !.' 
(15.5)
The kinetic energy of the electron is A* = £mr*. If, convention
ally, we take K p — when the electron is far from the nucleus, the
potential energy /*.'„ of the electron in orbit is
E =
p (4irt«)r
so its total energy is
E = A* + K„  Smv ! 
(Smb>
from which Eq. (15.5) gives
I s" e»
/; =
2 (4jre«)r (lvtn)r
1 c 2
2 (4Te )r
(15.6)
(15.7)
(15.H)
Of course, the kinetic energy of the electron is positive, but its
total energy in a stable orbit is negative since it is bound to the
nucleus, and work equal to /i' must be supplied to remove the
electron from the atom (process of ionization), Fig. 15.4.
Bohr's third assumption says that the permitted values of
electron angular momentum are
mm = n rr n = ' i % 3,
4/W
(15.9)
The radii of permitted orbits are obtained by solving Kqs. (15.5)
and (15.9) for the quantity (mr) 2 :
(mv)i  jEsp and (m » )s
210 Looking In : Atomic and Nuclear Physics
and equating the results to got
 n'
n _ 1, 2, 3, . . .
4jr*me ! l^me 4
From Eq. (15.8), the total energy can he written
(15.10)
Sn = 
I
2 (4we D )r
(4jreo)W
n = 1, 2, 3, . . . (15.11)
These are the only energy levels possible for the hydrogen atom in
the Bohr theory. The energy values for levels I to arc indicated
t
Totcl energy
E
•:• UnquanHzed
O
Radius r — —
__ f
Tj
i
""■""■■s^Binding energy
r 3
y
l\'
/
^^. ,
Ionization
P=  — *\
MM
srgy
1 +
Fig. 15,4 Bohr's model of the hydrogen otom.
in Table 15.1. The frequencies of radiation which the atom can
emit or absorb are predicted from Eqs. (15.3) and (15.1 1) as
A',  E, 2tt'W / 1 1 \
(15.12)
Hydrogen Atom— Bohr Model 211
Mini
1 = " _ 2 * im < A (A L\
X c (4rco)%^e \n/ n, s /
(15.13)
li =
= 1.097 X 10 s A>
from which the value of the Rydberg constant R can be verified as
2xW
(■iirtn)''/r'r
The orbit for which n = 1 is referred to as the lowest state,
the ground state, or the normal state for the hydrogen atom. If an
electron in the lowest energy state receives 12.07 ev of energy by
collision with an electron or by absorption of a photon, it can be
"kicked up" into energy level n = 3 (see Table 15.1). The time
interval before the electron spontaneously drops back to a lower
energy level is called the lifetime of the excited energy state, and
is ordinarily about 10 s sec. The electron we. are considering
might drop Brat from state n = 3 to « ■ 2, then from «. = 2 to
n  1. It would thus be responsible for the emission of two
photons. One would have the frequency of the first (H„) line in
the Balnier series; the other would contribute to the first line in
the Lyman series.
The ionization potential is defined as the energy needed to
remove, from an atom an electron initially in the lowest energy
state. The lesser energy needed to promote an electron from one
state to another of greater energy is called an excitation potential.
Obviously the hydrogen atom lias only one ionization potential,
but several excitation potentials. An atom with many electrons
lias a corresponding number of ionization potentials. Because of
the Pauli exclusion principle (Chap. 10), only in II and He do all
the electrons have it = I in the ground state. In other atoms, the
ground state is taken as the state of lowest energy.
15.4 EXTENSION TO HYDROGENLIKE IONS
Bohr's model and theory apply successfully to ions which have
only one electron, that is, I!e + , Li ++ , Be s+ , etc. The equation for
the Coulomb force is modified to read F = e(Ze) / (■iwuijr* where
Ze is the charge on the nucleus. This leads to inclusion of Z s in
Eq. (15.13), and thus
IWL JL\ (15.14)
X \nr my
212 Looking In: Atomic and Nuclear Physios
Hydrogen Atom— Bohr Model 213
This relation predicts that He + (Z = 2) should radiate a series of
iiiies hi the visible region for transitions to «/ = 4 similar to the
Balmer series for H (Fig. 15.5). This Pickering series for Hc + was
He*
*t
■4 
■3 
— 8:
— 6
Fig. 15.5 Comparison of energy
levels for H and for He 1 .
n = \
n = 2
observed first in star spectra and was subsequently identified witii
a laboratory helium light source.
15.5 CORRECTION FOR CENTEROFMASS ROTATION
The frequencies in the Pickering series for He + are not precisely
the same as those in the Maimer series for H, as Bq. (15.14) pre
dicts. Also, the heavy isotope II* has spectrum lines slightly
shifted in frequency from those which Kq. (la. t:i) predicts should
M
CM
Fig. 15,6 Rotation about center of mass (CM).
be identical for both H 1 and TI. These discrepancies suggest thai
instead of simply considering that llic electron moves around a
fixed nucleus, we should consider that both electron and nucleus
move about their common center of mass (Fig. 15.6). Let r f and
r„ be the distances from the center of mass to the electron and to
the nucleus, respectively. Then r = r e + r„ is the distance.
between electron and nucleus. If we introduce the angular
velocity w  »/r, [Kq. (15.5)1, for the centripetal force
mi
(4r tu )r :
becomes nr \ « = mr *>° (18.15)
tMreojr
The equation quantizing angular momentum, Eq. (15.9), becomes
mr,*a> + .l/r„ 2 u = n 7 r
2ir
(15.16)
where .1/ is the mass of the nucleus. From the definition of center
of mass,
M
m + ,1/
and
r„ =
m
m + M
By combining Eqs. (15.15) and (15.17), we find
(4^jH = m ">* rwi
and
!•'*,.< =
w m irw ■"fts
2.TT
where the symbol »i r ..,i is used for the "reduced mass,"
m\! m
Wr,.i = ;
m + .1/ 1 + tn/M
(15.17)
(15.18)
( 1 5. fit)
< 1 5. JO.
Since Fqs. (15.18) and (15.19) differ from Eqs, (15.15) and (15.16)
for no nuclear motion only in the replacement of electron mass m
by reduced mass »w, we see that the energy /■.'. corrected for
motion of the nucleus is related to the uncorrected energy A'„ of
Eq. (15.11) by
B. = m "" /■:„ 
m
E„
1 + w M
(»«)*
(15.21)
since m/M « 1 .
If we use the corrected expression for energy in Eq. (15.11),
the Bohr equation (15.13) for wave numbers becomes
1 u _2irV_ mM f \_ I \
X (47r e „} a A 3 c m + M \rf nf)
(15.22)
214 Looking in: Atomic and Nuclear Physics
This correction shifts each energy level by ahout 0.055 per cent
for H 1 . For the isotope II 1 , the shift is less. Hence a frequency
difference can be observed when two isotopes of an element are
present in a light source. The first (II a ) line hi the Balmer series
of II 1 has wavelength (in(>2.80 A; that for II 2 lias wavelength
6561.01 A. The reduced mass correction also explains why the
energy difference, and hence the frequency of radiation, is slightly
greater for the helium ion He + (say, from n = 6 to n = 4) than
for the corresponding transitions (n = 3 to n = 2) for hydrogen.
15.6 THE CORRESPONDENCE PRINCIPLE
Bohr's correspondence principle is the guiding idea that, in the
limit, the laws of quantum theory must join and agree with
classical theory (which does not involve It). This asymptotic
approach is to be expected when we go from microscopic systems
to those of larger dimensions, or for large values of the quantum
number n.
Bohr's theory for the hydrogen atom does show such agree
ment. With the aid of Kqs. (15.5) and (15. 10), we may express the
frequency of rotation of an electron in a Bohr orbit as
f'orb — tj~ —
v
2Hr
I
C4
ire u )mr
■]' =
me*
WllW
(15.23)
On classical (Maxwell) theory we should expect this electron to
radiate energy of this frequency, and possibly its harmonics.
But the theory which includes Bohr's quantum assumptions
for the 11 atom gives for the frequency radiated
me* (I _1_\
Now
ii/ 2
nr
nf
«,'
«,%/
(n, 4 n,)(nj  n f )
If rii and n; are both large compared to 1 and if An is small, we
can write this approximation
nf 1
n, s
2n Am
2An
n 8
Hydrogen Atom — Bohr Model 215
where An = tii ~ n/ and n»?ij« %. Then the Bohr frequency
becomes
Vln.lir ~
me*
4« 2 A s a 3
An
(15.24)
Comparison of Eqs. (15.24) and (15.23) shows that for large
orbits (large n) and for Are = 1 , the atom radiates the frequency
expected from classical electromagnetic theory. For An > 1, we
get harmonics. This is an example of a transition region between
macroscopic and microscopic physics where the laws of classical
physics and quantum physics overlap.
QUESTIONS AND PROBLEMS
1. Which of the experimental observations mentioned in Sec. 15.2
are satisfactorily explained by the Bohr theory of the H atom? Are any
not explained?
2. How can the fact that the spectrum lines of hydrogen are sharp
be used to support the statement that all electrons have identically the
same charge ef
3. At what temperature will the mean kinetic energy of hydrogen
atoms be just sufficient to excite the H a line? Am. 93,40G°K
4. Selig Hecht showed experimentally that a darkadapted human
eye experiences the sensation of light when the retina is irradiated by as
little as 10 X 10~ 12 erg. What is the minimum number of quanta of
yellow light (5,893 A) which the eye can detect? Arts, about 3
5. Assume that a free electron having kinetic energy 24.2 X I0 so
joule unites with a H + ion, goes to the lowest (n = 1) level, and gives
up its energy in a single photon. What is the frequency of the photon
radiated? Ans. 36 X 10'Vsec
6. How much energy is there in a quantum of violet light, wave
length 4,358 A? In a quantum of yellow light, wavelength 5,893 A?
Ans. 2.84 ev, 2.10 ev
16
Quantum
Dynamics
In this paper 1 am going to attempt to
find the foundation for a mechanics of
quantum theory. This mechanics is
based exclusively on relations be
tween quantities which are observable
in principle (e.g., frequencies and in
tensities of line spectra, and not elec
tron orbits), ... W, Heisenberg,
1925
Beginning with Bohr's initial formulation of the quantum theory
of atomic structure in 1914. physicists recognized that the
mechanics of systems of atomic dimensions must obey laws differ
ent from the larger systems successfully described by the classical
mechanics of Newton. By 11)24, a new method of treating atomic
phenomena began to be developed. It is known as quantum
mechanics, quantum dynamics, or wave mechanics. The names of
L. de Broglie, K. Sell nidi tiger. W. Heisenberg, P. A. M. Dime,
ami EL U. Condon arc chiefly associated with this development.
The concepts discussed in this chapter bring us to an acecpt
216
Quantum Dynamics 217
able theory of atomic physics. We arrive at a logical branching
point in our path. Armed with a successfully tested theory of the
atom, we can now (I) try to understand and predict properties of
atoms in intimate aggregation (solidstate physics) or (2) we can
turn to investigation of the internal structure of atoms. A goal of
such nuclear studies might be ultimately to manipulate nuclear
particles to our use, as a chemist manipulates atoms to create
molecules with desired proper! tee.
16.1 PARTICLES AND WAVES
Planck's derivation of the law for the energy distribution of
blackbody radiation (1900) first brought to light the particle
(quantum) aspect of electromagnetic radiation. Einstein strik
ingly established this viewpoint with his explanation of the photo
electric emission of electrons from solids (1905). Photons were
endowed with momentum {hv/c) by the Conipton effect (1924).
Also in 1924, Louis de Broglie, proceeding from relativity theory
and the observation that nature is symmetrical in many ways,
suggested that whenever there are particles witli momentum p,
their motion is associated with (or "guided by") a wave of
wavelength
V
(16.1)
The square of the amplitude of the de Broglie (matlcr) wave in a
given region is interpreted as being proportional to the prob
ability of finding the particle of momentum p in that region. In
de Broglie's hypothesis about waveparticle duality, an electro
magnetic wave tS the de Broglie wave for a photon, and proceeds
with speed c. The de Broglie waves for electrons, protons, neu
trons, etc., are That electromagnetic waves, but "matter waves,"
which travel with the speed of the particle.
We shall now discuss (I) a verification of these de Broglie
waves and (2) something about how their value at various
points in space may lie calculated.
Since the de Broglie equation predicted that 100ev electrons
should have wavelengths of about I A, it was suggested that the
wave nature of mailer might be tested in the same way that the
wave nature of x rays was first tested. A beam of electrons of
218 Looking In: Atomic and Nuclear Physics
appropriate energy could be directed onto a crystalline solid
(Fig. Hi. In). The atoms of the crystal form a threedimensional
array of diffracting ©enters for the de Broglie wave guiding the
electrons. There should be strong diffraction of electrons in
certain directions just as for the Bragg diffraction of x rays.
(a)
O
(c)
Fig. 16.1 (o) Davis son and Germer apparatus, {fa) Angular
distribution of secondary electrons, (c) Interpretation in terms
of Bragg reflection of electrons (refraction of rays has been
omitted).
This idea was tested by C. J. Davisson and L II. Germer
using 54ev electrons and a crystal of nickel (Kig. HUfe). The
emerge] i ( beam showed an intensify peak for 6 = 50°. The wave
length calculated From the Bragg equation turns out to be just
h/p for a 54ev electron. (The fact that electrons are observed at
other angles is attributed to secondary emission: Some incident
Quantum Dynamics 219
electrons collide, with and share their kinetic energy with some of
the electrons in the solid, with the result that some of these are
emitted at random angles.) Experiments on electron diffraction
confirms the hypothesis that their motion is directed by a wave of
some kind, and the wavelength agrees with that predicted by the
de Broglie relation, A = h/p.
16.2 DIFFRACTION OF PHOTONS AND NEUTRONS
De Bro^lie's hypothesis suggests that particles of any type may
exhibit diffraction effects. The diffraction of neutrons has been
useful in the investigation of the structure of solids. Beams of
neutrons whose wavelength is roughly equal to the spacing of
atoms in a solid can be obtained from a nuclear reactor. These
beams are diffracted by layers of atomic nuclei. On the other
hand, x rays are diffracted from planes in the solid where the
density of electrons is highest. Thus the two types of experiment
can give supplementary information about the structure of a
solid. Xray investigations reveal the location of the (bound)
electrons in a solid ; neutron diffraction reveals the arrangement of
the nuclei.
16.3 WAVE MECHANICS
We expect that a de Broglie wave will obey the same type of
secondorder differentia] equation (Appendix C) used to represent
other waves (Chaps. 9 and 12).
Important applications of Schrodiugor's equation are to cases
where the electron is subject to forces which hold it in a certain
region, as in an atom or in the atomic lattice of a metal. The
potential energy of the electron then varies from point to point.
As the simplest case of this type, let us examine the wave function
i>(x,i) for a particle of mass m which can move along a line
between stops a distance L apart, like a bead on a stretched wire.
Fig, 16.2 Particle confined to
linear motion within range L.
m
o
220 Looking In: Atomic and Nuclear Physics
The particle will never be outside the interval < x < L; so
^ is zero for x < mid x > L. Inside the region considered f here
are no forces on the particle; it is a free, particle. So the wave
equation has sine and cosine, solutions, hut these must he zero at
the. ends of the allowed interval. The allowed wavelengths of the
de Broglie wave arc A = 2L/n, which leads to
h nh
P = \ = 2L
(lfi.2)
showing that the linear momentum is quantized. The kinetic
energy of the particle is
p a I ii' ; h
k ~ 2m~ 2m 4JS
8mP
(18.3)
and since we have taken K v = 0, the total energy must have one
of flic values
The particle is located by the matter wave
iff = ( ^D sin  . J cos ait n = 1 , 2, 3, , . .
(164)
(16.5)
The amplitudes of the standing waves for states of motion cor
responding to n = 1, 2, 3, ... , vary as shown in fig. 1(5.3.
(There is a close analogy with standing waves in a vibrating
string.) We see that the act of localizing or hounding a particle
leads to the requirements that (1) the energy of the system can
take on only certain values and (2) zero is not a possible value of
the kinetic energy.
Another important type of prediction from wave mechanics
deals with the "leakage" of particles across an energy barrier.
Suppose we have a particle bound in a shallow potential energy
"hole" (1'ig. Hi. 4). There are now two kinds of solutions for the
wave equation. There are solutions for any K > 0. Particles in
these states have enough energy to escape; ^ extends over all
space for them. Hut for particles whose lit is less than #„n (Tig.
1G.4), the total energy is negative, and for K < the wave equa
Quantum Dynamics 221
tion has solutions for only certain values of the energy. The
higher the energy the more nodes there are in the wave (Fig.
16.46). The solutions are sinusoidal inside the well and have
exponential tails outside. Thus (here is some probability for lind
Fig. 16,3 Wave functions for a bead on
a string, for states n = 1,2,3,4.
'1 4
x— *
'.
t
>■■
'>i
(a)
X— \L
Fig. 16.4 (a) A "square" potential hole, and (b) the wave function of its states
n = 1,2,3.
ing particles in a region where, according to classical theory, they
do not have enough energy to be. Around a nucleus we may think
of a potential harrier whose craterlike shape is determined by the
Coulomb electrostatic force and a shorterrange 1 force of nuclear
222 Looking In: Atomic and Nucfear Physics
attraction. The wave viewpoint predicts that, charged particles
which do not have enough energy to go over the top of this
barrier have a small but not zero probability of occasionally
tunneling through the barrier.
16.4 BOHR ORBITS OR DE BROGLIE WAVES?
If we apply the concept of matter waves and the probability
interpretation of Sehrodinger's equation to the hydrogen atom,
we find that I he features which the Bohr theory correctly pre
dicted (only with the aid of arbitrary assumptions: mr = nh/'lir,
etc.) follow as a natural outcome of the mathematics involved.
The quantum dynamical treatment provides additional informa
tion as well. The electron in a hydrogen atom has potential
energy (■'■ \irt„r. If we write for the radial distance
r  Vt 2 + y* + z 1
Sehrodinger's relation Appendix C, Kq. (4} becomes
(16.fi)
The solution of this equation, ^(r,ij,z), is a function which has a
definite value at each point in the neighborhood of the nucleus.
To discuss this equation, it is convenient to change from reetan
Fig. 16.5 Rectangular and spherical co
ordinates.
Quantum Dynamics 223
gular coordinates to spherical polar coordinates, using the
relations
r = distance of point a from origin = y/x 2 + y* + z z
8 = angle from z axis to r = cos~ l {z/r)
4> = angle around z axis measured from x axis = tan 1 (</ as)
With the introduction of the coordinates r, Q, $, Kq. (Ifi.O) can be
separated into three ordinary differential equations, a fact we
represent by
f(r,e,<t>) = rt(r)0(0)*W
(lfi.7)
The function li describes how f varies as we go out from the
nucleus in a definite direction. The functions 8 and <$ describe
how ^ behaves from point to point on a sphere of radius r.
The equation for the function K(r) has a solution for any posi
tive value of E. These solutions correspond to states in which
the electron has enough energy to escape from the atom; there
are no quantum restrictions on the energy of a free electron. But
there are only certain negative values of E for which Kq. (16.7)
has any continuous solution. When the electron is bound in the
atom, an acceptable wave function ^ exists only if E has one of
the particular values
E„ =
me 4 1
I3.fi
ev
n = 1, 2, 3,
(10.8)
These are the same values for the energy states that the Bohr
theory predicted. The quantum number n is here related to the
part R(r) of the wave function which describes the probability
per unit volume of finding the electron in a given volume element
at various distances from the nucleus. This is independent of 8
and *. We can compute the average distance of the electron from
the nucleus by averaging over the probability distribution. The
result is roughly the same as the radius of the first Bohr orbit.
The energy (Kq. 10.8) is in exact agreement with the Bohr theory.
For each value of n, the equation for 8(8) is found to have one
or more solutions, described by a second quantum number I.
This quantum number takes on only the values
/ = 0, 1,2,3,
n  1
(16.9)
224 Looking In: Atomic and Nuclear Physics
Solutions of the * equation arc related to solutions of the R equa
tion such that the electron is less likely to he found near the
nucleus when in a high/ state than when in a low/ state of the
same energy.
For each value of /, the equation for *(<£) is found to have one
or more solutions, designated hy a third quantum number jtcj.
This takes on only the values
m , = /,_(/ i), (I 2),
— 10 12
(I  I), I (16.10)
Xo solutions of Schrodinger's equation for the hydrogen atom
exist for any other values of «, /, and m ( .
16.5 THE QUANTUMNUMBERS GAME
An atom can be completely described by the use of just four
quantum numbers for each electron. Three of these we have
already introduced.
The principal quantum number » determines the energy,
E<. (10.8). It may have the integral values n = 1, 2, :i, . . . .
The orbital angular momentum quantum number I deter
mines the angular momentum of the motion of the electron about
the nucleus. It may take on any integral value from to n — 1.
The corresponding value of the electron angular momentum i
VKI+ 1) h/2n.
The component of the orbital angular momentum along the
a axis is given by null, 2x, where m may lake on any of the 2/ j I
values: 0, ±1, ±2, . . . , ±1. The quantum number mt is called
the magnetic quantum number because physically the presence,
of an external magnetic field is necessary to establish a reference
direction (z axis) in space. In a magnetic field, the electron's
angular momentum is said to be "spacequantized" because its
component along the di recti >i the magnetic field is restricted
to the values mji 2tt (Fig. l(i.(i).
We now introduce a fourth quantum number 8, the electron
spin angularmomentum quantum number. This quantum num
ber defines the internal angular momentum {and associated mag
netic moment) which an electron is found to have, independent of
its orbital motion. An experiment to show this property of the
electron was performed by Stern and Clerlach. If a neutral atom
Quantum Dynamics 225
which has a magnetic moment passes through a uniform magnetic
field, it experiences a torque, but no deflecting force. If, however,
the field is nonuniform, the atom experiences a net deflecting
force as well. Consider a beam of II atoms. The electron in the
normal state has zero orbital angular momentum for n = 1,
/ = 0, mi = 0. There is no magnetic moment due to orbital
motion. Vet the beam of II atoms is observed to split into two
parts, each associated with a restricted orientation of the clec
Fig. 16.6 Possible orientations of angularmomentum
vectors.
tron's spin angular momentum. The two possible values of the
component of the spin angular momentum in the direction of the
magnetic field are ±%k/2*. We conclude that, unlike the other
quantum numbers, which are integers, n can have only the value
£, The component of the spin angular momentum may be either
parallel or antiparallel with the applied magnetic held. So we can
define a spin magnetic quantum number m„  ±\ and write the
component of the spin angular momentum in the direction of the
applied field as mji/2w.
(«>
J Traces on
receiving
I
Direction atoms move 1
Fig. 16.7 Magnetic field used in SternGerloch experi
ment, (a) With no field there is a single beam, (b) With
field, beam splits; some atoms are deflected toward N po!e,
some toward S pole. Traces where beam strikes detecting
plate are shown at top. (Adopted from R. 0. Rusk, "Afomk
and Nuclear Physics," Applet on Century Crofts, Inc., New
York, 1958.)
Quantum Dynamics 227
16.6 THE PAULI EXCLUSION PRINCIPLE
In 1925 W. Paul! suggested that a complete description of the
atom must include a unique description of each electron in the
atom. Xo two electrons in an atom may have identical values for
a set of four quantum numbers.
To see how this rule operates, consider the number of elec
trons permitted in the first orbital group (or shell) for which
n = 1. Since n is 1, 1 = and m t = 0, But m, may be +1 or  '..
So in this first group there may be two electrons, distinguished
only by having their spins in opposite directions. The continua
tion of this assigning of quantum numbers to electrons in many
electron atoms is shown in Table 16.1.
Table 16.1 Numbers of electrons in groups (or shells) as determined by Pauli's
exclusion principle
Orbital
group
n
/
m
s
No. elec. in No. elec. in
subgroup completed
group
1
{
1
1
i
2s 1 2
Iii the terminology of Table 16.1, we replace the term "orbit"
by "group" or "shell" (determined by n). This emphasizes the
threedimensional nature of the atom. The shells are often named
the K, L, M, . . , ,Q shells, corresponding ton =» 1,2,3, ... , 7.
226
228 Looking In: Atomic and Nuclear Physics
Within a shell, electrons with a common value of / form a sub
shell These arc designated s, p, rl, or / subshells according to
whether / has the value 0, 1,2, or 3,
16.7 BUILDING THE PERIODIC TABLE OF ELEMENTS
When the elements are arranged in order of increasing atomic
number, a periodicity in their chemical properties becomes
apparent, as shown by Mendeleev. The structure of the periodic
table is in agreement with the ideas of filled shells and subshells
as predicted by the 1'auli principle. We may "build up" an atom
by putting each electron in the shell of lowest energy until the
quota of permitted states is filled. Any additional elections must
be put. in the next shell as shown in Table lfi.2. The final column
of Table KS.2 is a description of the electrons in the outside shell
for the normal (ground) state of the atom. The electron configura
tion of an atom is described by the abbreviated not at inn ui" t he
last column. For example, % means there are two electrons in
the n  .i, I = I subshell.
The quantum numbers we are using were originated for the
case of one electron. J I is remarkable thai by assi g nin g occupied
states in terms of these numbers we get ati accurate description of
many of the properties of complex atoms. Kvidently the various
Table
16.2
Electron configured
ion for 1
ght atoms
>»,/*
Z
1,0
2,0
2,1
3,0
3,1
3,2
4,0
Configu
Element
(U)
(2s)
(2 P )
(3*)
(3 P )
(3d,
(4s)
ration
H
1
1
If
He
2
2
w
Li
3
2
I
2*
Be
4
2
2
2s*
B
5
2
2
1
2p
L
6
2
2
2
2p=
N
/
2
2
3
2p*
O
8
2
2
4
2p 4
F
9
2
2
5
2p»
Ne
10
2
2
6
2p r>
No
11
2
2
6
T
3s
Quantum Dynamics 229
electrons in a complex atom must disturb each other's orbits very
little.
One sort of disturbance, called screening, should be men
tioned. An outer electron is in a weak electric field because inner
electrons screen it from the positive charge of the nucleus. Hence
states in which the electron has some probability of being found
very near the nucleus will have lower energy (greater binding)
than those states in which the electron tends to stay outside the
screening inner electrons. Of the solutions of the Schrodinger
equation for a given n, those with lower values of J will tend to
/
2.1
16
12
B
4 
_i I 1 I i i i
■ !■■!
I ... I
10
I i 1
14
IS
22
26
30 X
Fig. 16.8 Variation of ionization energy (J in ev) with atomic number Z, sug
gesting greater stability of certain electron configurations.
penetrate the cloud of screening electrons most. Hence, for atoms
containing more than one electron, penetration causes the energy
of an orbit to depend on I. as well as on ti. (In terms of the liohr
picture, energy depends on the shape of the orbit as well as on its
size.)
Klectrical measurements which correlate well with electron
configurations are shown in I'ig. Ili.8. The ionization energy is
the work needed to remove the least tightly bound electron from
an atom. The variation of ionization energy with atomic number
7. suggests that certain electron configurations have relatively
great stability. The first is for helium, where the it = I shell has
230 Looking In: Atomic and Nuclear Physics
its ii"" 1 '' 1 <>f two electrons. The sharp drop to the binding energy
for lithium is attributed to the fact that the third electron must
bi> ndded io l lie n  2 shell and is therefore farther from the
nucleus. Tor the elements after lithium, there is a trend toward
increasing binding energy until another maximum is reached at
neon, when the n = 2 shell is filled. Like He, Xe is an inert gas.
This variation in binding energy is repeated several times in the
periodic table, each time giving a maximum binding at an inert
gas, followed by a minimum for the succeeding alkali metal. The
size of atoms also oscillates from shell to shell, about a value
approximately 1 A for the radius. In each shell, the alkali metal
has the largest radius.
16.8 CHARACTERISTIC XRAY SPECTRA
When a target is bombarded with electrons of high energy (l>),
x rays are produced which have a spectrum which is continuous
up to the maximum frequency given by the relation fcjWx = I <■■
In addition, xray spectrum lines arc observed at frequencies
which are characteristic of (determined by) the target material.
Characteristic xray spectra can now be explained in terms of the
shell structure of atoms. First, a vacancy must he created by the
displacement of an inner electron from, say, the K or L shell.
Since there are usually no nearlying vacant energy levels to
which these electrons may be promoted, they must be removed
altogether from the atom (ionization). This may he accomplished
when atoms of the target are bombarded by electrons which have
been accelerated through a potential difference of many thousand
electron volts or by highfrequency photons. Transition of a near
lying electron then occurs to Jill the vacancy. If a vacancy in the
K shell is filled by an electron from the h shell, an xray photon is
radiated whose frequency depends on the difference in energy
between the K and L shells. The vacancy left in the L shell is in
turn tilled by an electron from a still higher energy state, with
radiation of a photon of somewhat lower frequency, Fig. lfi.9.
Since the energy of the electron in the K shell is chiefly
determined by the nuclear charge Z, Moseley found he could use
the K a lines of the elements to identify the atoms in the target of
the xray tube. lie found a linear relation between the square
root of the frequency and (Z — l), as would be expected from
Quantum Dynamics 231
the Bohr formula with allowance for screening by the inner K
electrons. From the relation
= Rc(z  ]y(L L\
(16.11)
Moseley was able to prove that early assignment of atomic
numbers to cobalt and nickel was in error. The atomic mass of a
natural mixture of the isotopes of Xi is 58.(i!) and for Co 58.94.
They were first placed in the periodic table in the order of increas
ing atomic mass. Hut the xray lines showed that this order
should be reversed, for Z Co = 27 and Zy, = 28. Moseley 's work
K
0.2 0.4 0.6 0.8 1.0
Wavelength, A
(a)
E,
E M
(b)
K«
K »
L a
H
\ <M a
M
N
Fig. 16.9 Characteristic xray spectrum, (a) Molybdenum target with V
(a) Simplified energy level diagram.
35 kv.
(1913) gave the first accurate method for measuring atomic
number, Z. The committee awarding the 1917 Xohel Prize to
C. G. Barkla, for his work on characteristic x rays, stated that
Moseley would have shared the award but for his death at
Callipoli.
16.9 PHYSICS OF THE SOLID STATE
Our theory, based on the nuclear atom model and quantum
mechanics, tells us that under ordinary circumstances of tem
perature and pressure the nuclei of atoms will never get very close
to one another. The combination of atoms should therefore be
232 Looking In: Atomic and Nuclear Physics
explainable through the exchange or sharing of electrons. In
terms of tfic measured masses and charges it should be possible
to describe the formation of molecules and chemical reactions.
One might also hope to describe crystal lattices and the mechan
ical, thermal, electric, and magnetic properties of solids. Prac
tically, the difficulty is the complexity of the computations. We
shall examine some of the successes of quantum mechanics in
explaining important electric and magnetic properties of solids.
This comprises hut one segment of solidstate physics in which
there is very active research.
16.10 CLASSICAL THEORY OF CONDUCTION IN METALS
A theory proposed by Drude and Lorentz, soon after the dis
covery of the election, assumed, as have later theories, that some
of the electrons are free to travel throughout the whole volume of
a crystalline material. In a "good" metal, it was assumed thai
there is about one free electron per atom and that the number of
conduction electrons is independent of temperature. These elec
trons dart around in all directions ui) h the high speeds of thermal
agitation. Hut if an electric held is applied, the "electron atmos
phere" experiences a relatively slow drift, superposed on the
random thermal motions. The electron drift is the electric current.
The transfer of any increase in the energy of random motion in
any direction constitutes thermal conduction. To make quantita
tive predictions, it is necessary to make some assumptions about
the distribution of electron speeds. Theories have differed in
these assumptions.
The classical theory assumed that the electron speeds followed
the same distribution law as .Maxwell and Bollzmanii had used
for molecular speeds in developing a successful kinetic theory of
gases (Chap. 8). Among a large number N of electrons, the frac
tional number N,'N having speed r is given by
AT vV \2fc27
(16.12)
Tf we plot this expression against o, the area under the curve
between r, and v> equals the fraction of all the electrons whose
speeds are between r, and r*. Since kinetic energy depends on the
Quantum Dynamics 233
speed squared, the average kinetic energy depends on the average
of the squares of the speeds. The square root of this average is
called the rootnieansquare (rms) speed. The distribution curve
becomes flatter and the maximum shifts toward higher speeds as
the temperature increases (Chap. 8).
The classical theory gives rough predictions of the electrical
and thermal conductivities of metals, if is in accord with the
experimental observation that the best conductors of electricity
are also the best conductors of heat. Wiedemann and Franz ( 1 850}
showed that the electrical conductivity thermal conductivity
ratio is a constant, for metals. The classical theory, using known
values for e and k, predicts that the thermal conductivity/
electrical conductivity ratio = li.ll X 10 _B 'T cal ohm sec (°K).
Fig. 16.10 Hall effeel
D D'
—n —
This checks well with values measured for platinum and other
I mre metals.
But the classical theory meets with significant failures. It
predicts that the free electrons should contribute /f to the
specific heat of a crystal. This considerable electronic specific heat
is not observed experimentally. Also, the theory is unable to
explain the enormous range of electrical resistivity for different
materials. Further, the theory suggests that since the free elec
trons have magnetic moments, even a weak magnetic field should
produce a large paramagnetic magnetization (magnetic moment
per unit volume) in a conductor. It does not. Finally, the theory
has difficulty in predicting the sign of the Hall coefficient. For a
currentcarrying conductor (Fig. Hi. 10) one would expect that a
potentiometer connected between Cand J), in a plane perpendicu
lar to the current . would indicate zero potential difference. If now
an external field H is applied, the conduction electrons experience
a magnetic thrust perpendicular both to H and their velocity v.
The equipotential line CD is tilted through some angle <j> to
position CD'. The classical theory predicts that tan 4> (the Hall
coefficient) should have the same sign for all metals. It docs not.
234 Looking In: Atomic and Nuclear Physics
16.11 FREEELECTRON QUANTUM THEORY OF CONDUCTION
Fermi introduced a radically different description of the free
electrons iii a metal. He incorporated the exclusion principle,
assuming that the "free" electrons hi a metal are quantized and
that no two can act exactly alike. Momenta are quantized; only
two electrons (having opposite spins} can have a given mo
mentum. As the temperature is lowered, electrons settle down hy
quantised slops to lower momentum values. But as a consequence
of the exclusion principle, some electrons wil! remain at mo
mentum values considerably above zero: thai is, linn will have
appreciable energy, even at absolute zero temperature. When the
N
f
J0°K
i
300° K
\Very high temp.
Fig. 16.11 Fermi distribution of speeds at various
temperatures.
temperature rises, only the electrons of highest momentum can
accept thermal energy and move to still higher momentum
values.
The Fermi distribution law is expressed by
N = h l e m, "' i!_A "'* r
1
(16.13)
where E,„ is the maximum energy an electron can have at 0°K. In
Fig. 16.11, the progressive rounding of the curve as temperature
increases represents the shift of some electrons to higher energies.
The Fermi distribution curve should be compared with the Max
well distribution (Chap. 8).
The ['ermi theory successfully accounts for the slight partici
pation of electrons in specific heats. In Fig. Hi. 12, the Fermi
distribution of energy is plotted. At 0°K all energy states are
Quantum Dynamics 235
occupied up to a certain maximum (Fig. 16.12a). At a higher
temperature some electrons in upper levels have been able to
accept energy ami move to still higher [ovate (Kg. [6.126). Btl1
owing to quantum restrictions, relatively few electrons have
participated in the temperature rise. The Fermi theory predicts
that electrons in a conductor should contribute roughly 1 per cent
of the amount predicted by the Maxwell theory, in agreement
with experiments in calorimetry.
The fact that all energy levels, up to a certain maximum, are
filled means that for every electron traveling to the right in a
metal there is another elect ion traveling toward the left. Thus all
electrical conduction in the metal must be due to the relatively
few electrons near the top of the distribution (Fig. JO. 126) which
Fig. 16.12 Fermi distribution of
energies, showing (a) all levels filled
up to a maximum of K, (b) some
electrons promoted to higher energy
levels at a high temperature.
Rel, no
Rel * no
can be excited easily to an unoccupied quantum level. One con
cludes that electricity must be conducted by only a small fraction
of the free electrons (rather than by all, as assumed in classical
theory). In turn, this implies that an electron must be able in
travel long distances without being bumped by ions in the crystal
lattice. The freeelectron quantum theory, like the classical
theory, is unable to account for the distinction between con
ductors and insulators.
16.12 BAND THEORY OF CONDUCTORS, SEMICONDUCTORS,
AND INSULATORS
In the modem band theory of the electronic structure of solids,
the effects of the lattice ions on the free electrons are considered
to explain the occurrence of conductors, insulators, and semi
conductors. The moving electrons are pictured in terms of
236 Looking In: Atomic and Nuclear Physics
do Broglie waves of wavelength \  h/mr. The influence of the
lattice ions arises from the variation of potential from atom to
atom in the crystal (Fig. Hi. 13). The passage of I he do Broglie
waves is treated mathematically by methods similar to those
used in investigating the passage of light waves through a similar
lattice.
Yinnrvv
Fig. 16,13 Variations of poten
tial along a onedimensional
crystal lattice.
It turns out that the graph of electron kinetic energy versus
momentum, instead of having the parabolic shape (Fig. Hi. 14a)
which it would have in a conductor where there was no variation
of potential, jumps discontiuuously for particular values of
de Broglie wavelengths (Fig. 16.14/*). Not all electron momenta
(a)
(b)
Fig. 16.14 Energy vs. momentum: (a) Assuming no variation of
potential between atoms; (b) assuming a variation of potential
similar to that of Fig. 16.13.
are possible. From this point of view, the effect of the ion lattice
is to preclude certain values of electron momentum and hence to
leave forbidden energy gaps at these momentum values.
The properties of conductors, insulators, and semiconductors
ran now be interpreted in terms of the conduction bands (Pig,
Hi. 15). If the highest energy band containing electrons is full and
Quantum Dynamics 237
is appreciably separated from other bands (a), the materia! is an
insulator. To produce a current in such a material, electrons have
to be advanced across an energy gap large compared to thermal
energy /,•'/'. In a conductor, however, the highest band containing
electrons is not full (/*). Kven a small external electric field can
Allowed (Empty)
Allowed [Partly full
Forbidden / £ w » « r
Forbidden
; (Allowed
i) (Full)
(a)
(6)
Allowed (empty)
Forbidden E 9 , p 2skT
Allowed (full)
(C)
Fig. 16.15 Distribution of electrons in bands in (a) an
insulator, (b) a conductor, and (c) a semiconductor.
produce an unbalanced momentum distribution {a current) by
promoting electrons to energy states of small excitation. Semi
conductors arc an intermediate case in which the highest occupied
hand is full (c), but. the energy jump to the next band is compar
able to kT. Increase in temperature would be expected to lower
the resistance of a semiconductor.
17
Radioactivity
The new discoveries made in physics
in the last few years, and the ideas
and potentialities suggested by them,
have had an effect upon the workers in
this subject akin to that produced
in literature by the Renaissance.
J, J, Thomson, in an address on
radioactivity, 1909
Radioactivity has provided us with much of the knowledge we
now have concerning the nucleus. Emission of a and ji particles
by certain atoms suggested the idea that atoms are built of
smaller units. Measurements of the scattering of a particles by
atoms confirmed Rutherford's idea of the nuclear atom. The dis
covery of isotopes can be traced to the analysis of the chemical
relationships among the various radioactive elements. The bom
bardment of atoms with energetic a particles from radioactive
su Instances was found to cause disintegration of some atomic
nuclei; this led in turn to the discovery of the neutron and to the
present theory of the makeup of the nucleus. The transmuted
atoms resulting from such bombardment are often radioactive.
238
Radioactivity 239
The decay of these artificial radioactive nuclides is in accord with
the laws found earlier in the study of natural radioactivity.
17.1 TYPES OF RADIOACTIVITY
In the theory of the nucleus there is no counterpart of the simple,
easily visualized mechanical model employed in the Bohr theory
of the atom. But the concept of energy levels, found so useful in
studying the atom, is carried over to the description of the
nucleus. Nuclear spectroscopy deals with the identification of
these energy levels and is an important source of information
about the nucleus, since radioactive changes can be measured
with high precision.
When the electronic structure of an atom acquires some extra
energy, the atom almost always gets rid of this extra energy very
quickly, returning to the ground state in roughly 10  * sec. It does
so by emitting one or more photons or an electron if there is
enough extra energy. Many nuclei, however, can exist for long
periods of time in an unstable state, that is, in a state from which
the nucleus can and eventually will decay to a stable state. A
nucleus may go to a state of lower energy by emitting an a
particle (a radioactivity), an electron or positron (fi radio
activity), or a photon {7 radioactivity).
Most "natural radioactivity" is found among the very heavy
elements (A > 210), which tend to be unstable to a decay.
These nuclei decay so slowly thai there are still some of them left
from the time of formation of the elements. Radioactive isotopes
not found in nature can lie prepared in nuclear reactions.
17.2 STATISTICAL LAW OF RADIOACTIVE DECAY
The activity of a radioactive sample is defined as the number of
disintegrations per second. The activity decreases with time.
Each radioactive isotope has its own characteristic rate of de
crease, figure 17.1 is the plot of the decay of a radioisotope which
decreases in activity by 50 per cent every 4.0 hr. The form of t In
experimental decay curve suggests that the decay is a loga
rithmic process. This is verified by plotting the logarithm of
activity versus time. A straight line results.
We can derive an exponential law of decay for a sample con
240 Looking In: Atomic and Nuclear Physics
tabling a large number of radioactive atoms. We assume that
each undeeayed nucleus has a definite probability X of under
going decay in the next second and that this probability is inde
pendent of time and is independent of whatever other atoms are
present. Then Iho number of decays in a time interval dt is equal
to the number of undeeayed atoms present times the probability
g
<
*m
c
3
%
^
fc
K T
">
\ t„=^L.
t
\ i " 0.693
<
1 i i i .i
4 8 12 16 20 24 t, hr
Fig, 17,1 Decrease in activity of a radioisotope
with a 4.0hr halflife.
X (ll that each one of them will decay. Thus the change (decrease)
in the number of undeeayed atoms is
(IN = \N(U
(17.1)
The decay constant X is the relative number dX/N of atoms
which decay per second. The value of X depends only on which
radioactive isotope we are considering,
Uy separating variables in Kq. {17.1), we obtain a simple
differential equation
tlX
N
Xdt
(17.2)
whose solution is
N = AV"*«
(17.3)
Radioactivity 241
where N a is the number of undeeayed atoms in the sample when
t = 0. (Note the mathematical similarity with the equation for
the exponential ahsorption of a beam of radiation.) The activity
of a sample, the number of decays per second, is given by
Activity = .: = XAV W
or
Activity = XN
(17.4)
(17.5.
The activity depends on the number of atoms present and on their
decay constant, X.
17.3 HALFLIFE
The halflife T of a radioactive substance is the time interval in
which the activity (and hence the number of undeeayed atoms)
decreases by 50 per cent. For the activity of Fig. 17.1 this is
4.0 hr. Itoiii the definition that t  T when X = IN^ Eq. (17.3)
becomes
hN a = AV" W
which gives
T = log, 2 = 0.«»3
(17.6)
(17.7)
The average life '/'„, or life expectancy, of a radioactive nucleus
may be calculated by Rimming the lives of all the nuclei and
dividing by the total number of nuclei
„, fi/ot dN I t  , A . . .. ., 1
? « = / 57 = tt / 'Wr M dt = r
Jo No No Jo X
(17.8)
The decay constant X is the reciprocal of the average life, in
accord with the interpretation of X as the probability of decay of
an atom per second.
17.4 UNITS OF RADIOACTIVITY
A unit of activity was historically defined as the amount of radon
(gas) in equilibrium with one gram of radium. The National
242 Looking In: Atomic and Nuclear Physics
Research Council in 1948 extended this definition to define one
curie as that quantity of any radioactive substance which gives
3.70 X 10 10 disintegrations per second. Since the curie is a rela
tively large unit, the millicurie (I mc = 0.001 curie) and the
microcurie (I ^c = 10 s curie) are widely used. A counter near a
radioactive source detects a certain fraction of the particles
emitted; the counting rate is proportional to the activity of the
source.
The specific activity of a radioactive source is the rate at
which 1 gin emits charged particles.
17.5 GAMMA DECAY
A nucleus in an excited statc(z*X i ) may go to a state of lower
energy by emitting the difference in energy as a photon:
zX* + hv C— M«v)
(1 7.! I )
Now y decay does not cause a change in the atomic number or
the mass number of the nucleus. The halflives for y decay are
seldom very long.
Study of y radiation gives important information about the
initial and final states of the nucleus undergoing a y transition.
Like the spectra of atoms, the y spectra of nuclei are found to
consist of sharp lines, showing that the nucleus has discrete
energy levels. The observed energies of emitted photons give
consistent results for the nuclear energy levels
hv  Ei  E } (17.10)
The electromagneticwave nature of y radiation is demon
strated experimentally by diffraction. This is feasible only for
those 7 rays of relatively low energy because ruled gratings or
crystals with effective spaeings about equal to very short y
wavelengths are not available.
The energies of highenergy 7 rays may be measured in
several ways. When a 7 ray ejects a photoelectron from the inner
shell of ati atom,
hv = E k + I
(17.11)
where E k is the kinetic energy of the ejected electron and / is the
binding energy of the shell from which it is removed. The ioniza
Radioactivity 243
tion energies (/) are known. Hence the energy of the yray
photons may be determined by measuring the energies of the
photoelectrons.
Positronelectron pairs (Chap. 19) can be created by 7 rays
with hv > 2m c'. The photon energy is transformed thus:
hv = 2m,e* + E k + + E k ~ + E k . tceoil
(17.12.
From conservation of momentum, the recoil velocity of the
nearby nucleus should be small. Its energy can generally be
neglected. Measurements of the momenta of the electron and
positron in a magnetic field then give information from which the
energy of the X ray can he found.
17.6 ALPHA DECAY
When an a particle is ejected from the nucleus, the original
nucleus loses two protons and two neutrons. Its mass number
decreases by four units while its atomic number Z decreases by
two. « decay thus causes transmutation of the parent chemical
element into a different chemical element
Z X* » «_,*** + .He' + Q (energy)
17.1:;.
Now a decay occurs spontaneously, without any external forces,
and it provides kinetic energy (#*,„) for the ejected n part ideas
well as some kinetic energy {E k , d ) for the recoil ''daughter" nucleus.
Hence a decay cannot occur unless the total rest mass decreases.
The decrease in rest energy is equal to the kinetic energy released,
called the disintegration energy Q:
Q = E k ,,i + E k ,„ = (m„  m lt  w Q )c !
(17.14)
To predict whether a nucleus will undergo a decay, we may com
pare its rest mass with the sum of the masses of the product
nuclei. Actually we can use the masses of atoms instead of those
of the nuclei. The same number of electrons are associated with
the initial and final nuclei, so the electron masses cancel in the
calculation of Q. From Eq, (17.13),
Q = (m*  mr — trine)*?
(17.16)
244 Looking In: Atomic and Nuclear Physics
Exampk, Find the Q value for the disintegration t t>N'l u * —* tlic* 4 S8 Ce I4l >.
Prom tables of isotope mosses:
♦He 4 = 4.00387
» a Ce l4 ° = 139.01977
143.95364
toX'!'*'  [43.95550
Produeta  143.95364
m = 0.00192
Q = mc* = 1.79 Mev
Example. In a decay, what fraction of the disintegration energy appears
as kinetic energy of (he a particle?
Conservation of energy and conservation of momentum in a decay
require
Q = Et.4 + E kia = lmj>S + £fli a p a *
m a v a = m&t
From (he momentum equation, v d  (»ia/»ij)e„. Substituting this in the
energy equal ion. ive have
or
Q
" ^'(S)
+ 1
AY~ =
a
1 + m„/m d
(17.16)
(17.17)
If .4 is the mass number of the parent nucleus, then m a /m a =* 4/(A — 4)
and
**„
44
(17.1 7«)
Thus for large .4, the a particle gets most, but not quite all, of the dis
integration energy.
An interesting feature of a decay called the tunnel effect may
be illustrated by data for a particular ease. One can perform an
experiment similar to the KutlierfordCeigerMarsden scattering
experiment (Chap. 15) using a thin foil of 94 U i!8 to scatter the
7.68 Mev a particles from mIV 4 (also called Ra C). One finds
that the Rutherford scattering law is obeyed. Kvtdently the
a particles from IV 4 do not have sufficient energy to get over
the Coulomb barrier; they are scattered away from the l"' :,s
nucleus. This is suggested in Fig. 17.2, which shows the potential
Radioactivity 245
/ u" 8
t *
n,
t
V,
7.68
1 Decay
4.20 Me>
1
Mev
»
/
r
Fig. 17.2 Coulomb borrier: scattering of a high
energy particle and tunneling of a lowenergy
particle.
Fig. 17.3 Wave mechanical description of tunnel effect.
energy curve of an a particle near a U m nucleus and a IV 4
a particle being turned away by the potential barrier. Contrast
this with the following fact: U 2,s itself is an a emitter, emitting
a particles whose kinetic energy is only 4.20 Mev. We have a
paradoxical situation : The lowerenergy U S39 a particle can cross
a barrier which the higherenergy I'o 414 a particles appear unable
246 Looking In: Atomic and Nuclear Physics
to cross. An explanation on the basis of classical physics is
impossible.
The wave nature of the a particle must be taken into account.
When we use wave mechanics to describe an a particle in the
nucleus, we find that a little of the wave function will "leak"
through the barrier so that there is a small probability that the
particle may be found outside (Fig. 17.8). According to wave
mechanics, if the a particle has enough energy to be outside,
then there is some probability that it will be found there. This
probability is very small for U 2M and accounts, roughly, for the
U 23 " 1 halflife of 4,f> billion years. The tunnel effect works in either
direction, so some of the IV" a particles used in the scattering
experiment must have penetrated the nucleus, but the fraction
which succeeded was negligible. The probability of tunneling
depends strongly on the height and width of the potential barrier.
17.7 BETA DECAY
The /3 particles emitted from a radioactive source are shown by
deflection experiments to be highenergy electrons. There are
good reasons to believe that, these electrons do not exist in the
nucleus but are created by a rearrangement of the nucleus into a
state of lower energy. Any excess of energy over thai required to
provide one electron rest mass (m.c) appeal's as kinetic energy of
the emitted electron.
An argument against the existence of electrons in a nucleus,
prior to emission, makes use of the uncertainty principle. If an
electron were confined in a region of dimensions no larger than
about 2r = 1.4 X 10 _ " tn, the electron would have momenta as
high as
Ap = = 3.8 X 10 ' kgm/sec = Um<c
■ttt &,v
and hence kinetic energy as high as
E* = \/{Ap')*c s + m<c*  m.c' 1 = 14m,c s = 7.2 Mev
It seems unlikely that there are attractive forces in a nucleus
which are sufficiently strong to bind an electron having this much
energy.
Radioactivity 247
Two different types of decay occur: ff~ decay, in which an
electron is emitted from the nucleus, and (3 + decay, in which a
positron is emitted. If the nucleus consists of neutrons and pro
tons only and if electric charge is conserved, then upon emission
of an electron, a neutron must be converted to a proton, hZ = f 1.
Similarly, positron emission involves the conversion of a proton
to a neutron, AZ = — 1.
(17.18)
(17.19)
For £~ decay to occur, the mass of the decaying nucleus must be
greater than the mass of the product nucleus plus the mass of an
electron. An atom which is heavier than the atom with Z one
unit greater but with the same .1 will decay into that atom by
0 emission.
The condition for £+ decay is slightly more complicated.
Q — nix — wiy — '2m r c 3
(17.20)
where m\ and m\ are the masses of the initial and final atoms,
respectively, and hi. is the rest mass of an electron. An atom is
ftf + unstable if it is more than two electron masses heavier than
the atom with the same .1 and one less Z.
There is still a third ($ decay process whose overall result is
the same as /J + decay. A nucleus may absorb one of its orbital
electrons. This process is called A" capture since the elect Tons ill
the nearest (re = 1) shell are most likely to be absorbed. The
energy rule is the same as that for 0 decay: If the resulting atom
is lighter than the original atom, it is unstable to K capture.
The changes resulting from various nuclear processes are
often represented in a protonneutron diagram (Fig. 17.4) in
which each nucleus is plotted in terms of the number (Z) of its
protons versus the number (A — Z) of its neutrons. It is a result,
of the processes we have just discussed that no two adjacent
isobars (nuclei with same mass number) can both be stable.
The heavier will #decay into the lighter.
The energies of electrons and positrons from decay have
been determined with various types of /3ray spectrometers. In
principle, they measure the momentum of an electron by (hiding
the curvature of its path in a known magnetic field. It is found
that electrons in a given type of 8 decay may have any energy up
248 Looking In: Atomic and Nuclear Physics
&T,
~
Orig.
nucleus
K CQpt ,
AZ
Rg. 17.4 A protonneutron diagram.
to the calculated energy release Q (Fig. 17.5), Here is a difficult;
with the hypothesis that & decay consists of the emission of an
electron (or positron) and the conversion of a neutron to a proton
(or proton to a neutron). Tor the nuclear change is from one state
of definite energy to another state of definite energy. Yet the
electrons emitted carry varying amounts of energy, up to the
maximum available. There is another difficulty. Consider the
# decay of a nucleus containing an even number of nucleons. Its
angularmomentum quantum number is an integer, since there is
an even number of spini particles present. If a single electron is
Fig. 17.5 A continuous spectrum.
now created, there will be an odd number of spin£ particles and
the total angularmomentum quantum number will be half an
odd integer. But a spontaneous change in angular momentum is
not possible.
To remove these difficulties, we assume that along with the
electron, another particle, also of spin4, is created and emitted,
Radioactivity 249
but not observed! This particle is called the neutrino. Since it
shares the disintegration energy Q with the electron, the con
tinuous energy distribution observed for the ff particles (Fig. 17.5)
can be explained. The neutron is assumed to have zero rest mass,
so the only change needed in our previous equations is to replace
E k by E k + SfcneutrUo The neutrino participates only in reac
tions. Since it has no rest mass, it travels with the speed of light.
It is postulated to have spin ■$ and to obey Pauli's exclusion
principle. The neutrino lias no electric charge, and it is difficult to
detect! This remarkable particle has been assumed as necessary
by physicists since about 1934. Its existence was first experi
mentally demonstrated in 1956, by detection of y rays produced
in a planned sequence of events initiated by the neutrino.
17.8 NATURAL RADIOACTIVE SERIES
In experiments which followed the discovery of radioactivity,
quite a number of substances were found to show activity. It was
found that certain of these substances were associated with each
other in series, the successive members being formed by the dis
integration of the preceding member, until a stable nucleus is
reached.
One can predict that there should exist four separate decay
chains or radioactive series. A nucleus belongs to one of four
classes, depending on whether its mass number A has the form
4n, 4n + 1, 4» + 2, or 4n + 3, where n is an integer. Radioactive
decay of a nucleus in one of these will result in the formation
of daughter nuclei in the same class. This follows since there
is no change in mass number in decay or in 7 decay, while
in a decay, A.l = 4. The four radioactive series are represented
in Fig. 17.6. Each bears the name of its longestlived ele
ment. The neptunium series is not observed naturally because
gaNp ! " (T = 2.2 X 10 s year) has almost completely decayed
since tin formation of the elements (about X 10' years tigo).
The decay schemes of these four series end with stable isotopes
of lead. A few radioactive isotopes which do not belong to the
heavyelement chains are found in nature, Table 17.1.
When the elements in a radioactive series are allowed to
accumulate, a steady state will be reached (if the parent atom has
a long halflife) in which the number Nx\i of atoms of one isotope
250 Looking In: Atomic and Nuclear Physics
NAZ
Thorium series
(A = 4n)
?
z,
>Th" J 
140
Ra
Ht (
V A
Lrv."*
Th
14
'Ho
/• '
130
Ph r;
Tl
Pb
!0S
N = AZ
140
Neptunium series
(A4H + I) Np J "
130
\ \ y\
1
R
Li»«'
Jf
Ml
r
J"V
/■A
Pc s "
Pb*"^.
r"
Tl™
1
N=AZ
80 84 88 93 Z
Uranium series
(A = An + 2) yJM
SO 84 88 92 Z
Actinium series
(A = 4n + 3)
6
u ;
j»
140
Th iJ '
131
Ac'V
1
* r itoJ
Th' 37
C 111
Rr,*"
130
I s "
Pc
15
Pb ,u
/
1 .
X
11
Hb
80 84 88 92 Z
80 84 88 92 Z
Fig. 17.6 Decoy schemes of the four families of natural radioactivity.
which decay per unit time is equal to the number A/oX* of atoms
of the* next isotope which decay per unit time, or
JV.Xi = N 2 \2  iVaXa = ■ ■ ■ (equilibrium} (17.21)
This equilibrium equation is often used to calculate X for an
Radioactivity 251
Table 17,1 Isolated natural radioisotopes
Isotope
Decay
Halflife (years)
isotope whose halflife is too large or too small to make a particle
counting experiment convenient.
PROBLEMS
1. Radium E has a halflife of 5.0 days. Radium E emits a 0
partiele to become radium P. (a) Which nucleus (E or F) has the greater
positive charge? (b) Starting with 1.0 gm of radium E, how long would
it take for J gm to decay into radium F?
2. a particles shot vertically upward arc deflected by the earth's
magnetic field in which direction?
3. Calculate the mass of Au l,s (7* = 2.7 days) in a source of 1.0 mc.
Ans. 4.0!) X 10" gm
4. Five mg of IV '" {T  140 days) are allowed to decay for 1.0
year. What is the activity of the sample at Hie end of that time?
Ahx. 1.35 X 10" disintegrations per second
5. A sample of radioactive sodium (Xa 1 . T = 14.8 hr) is assayed
at 95 mc. It is administered to a patient 48 hr later. What is the activity
at that time? Ans. 10 mc
6. What is the volume of 1.0 mc of radon. M Hn m (T  3.82 days),
at 0°C and I atm pressure? Ans. fi.fi X 10~" m !
7. Suggest a method for using data on the uraniumdecay series to
estimate the age of the earth, Suggest B waj of using the radioactive
isotope of carbon C u (T = 5,600 years) to substantiate the age of cot
ton fabrics found in an Egyptian tomb.
iH 3
r
12.4 j Created continuously
> by cosmic radiation
.C"
r
5,590 J in atmosphere
nK««
§,K
1.2 X 10"
^Rb"
r
6.2 X 10">
».lo ,li
r
6 X 10 14
stLo""
r
2 X 10"
M Sm 147
a
1.5 X 10"
7ito 176
r
2.4 X 10'°
76 Re"<
r
4 X 10 12
18
Nuclear
Reactions
No man will ever comprehend the
real secret of the difference between
the ancient world and our present
time, unless he has learned to see the
difference which the late develop
ment of physical science has made
between the thought of this day and
the thought of that. T. H. Huxley
A particle directed at a nucleus may undergo a collision (elastic
scattering) which leaves the struck nucleus unaffected. A second
possibility is that a nuclear reaction takes place producing sonic
change in the struck nucleus. The incident particle may be
absorbed into the struck nucleus. A rearrangement may occur in
which the incident particle remains in the nucleus and another
particle emerges. The incident particle may emerge but leave the
nucleus in a different energy state. There are other possibilities.
Nuclear reactions may be caused by individual nucleons, photons,
deuterons, a particles, and heavier particles.
252
Nuclear Reactions
253
The first artificial nuclear transformation was achieved by
Rutherford in 1919, in bombarding nitrogen with a particles
from a natural radioactive source, Ha C Because of the impor
tance of neutrons in nuclear reactions, we shall depart from
historic sequence to discuss first the discovery of the neutron by
Chadwick in 1932.
Among the achievements of nuclear studies are the production
of scores of valuable isotopes, the discovery of the neutron and
other particles, and the release of energy in the processes of
nuclear lis.sion and fusion.
18.1 DISCOVERY OF THE NEUTRON
Bothe (1930) found that when a particles from polonium fell on
a beryllium foil, a penetrating radiation was emitted. Irene and
Frederic Joliot observed (1931) that the intensity of this radiation
was apparently increased by passage through paraffin. They sug
gested that Bothe's radiation was y radiation which knocked out
fast protons from paraffin and other hydrogenrich substances.
Chadwick (1932) applied the equations for the Compton
effect to the headon collision of the assumed y ray and proton
(mass m) and showed that the maximum energy given to the
proton by a photon (hv) would be 2hv/(2 + mc*/hv). Experi
mentally the recoil protons from paraffin were found to have a
maximum energy of 5.7 Mev, requiring thai the y ray from Be
have energy hv ")."> Mev. Hut when aifcrogen was subotitvted
for paraffin as a target, the i.'2Mev recoil nitrogen ions which
were observed required that the same y ray have an energy of
90 Mev. Chadwick resolved this contradiction by suggesting that
the "rays" from Be were actually neutrons, whose existence had
been proposed by Rutherford in his mode! of the nuclear atom.
The fact that atomic masses (beyond i\V) are roughly twice
the atomic number suggests that the two types of particle
neutron and proton which constitute a nucleus have approxi
mately equal mass. Chadwick confirmed this expectation by
calculations made on the reaction
5 B" + a IIe<> ,»' + 7 N M + Q
08.1)
Three of the four masses were known. The energy of the incoming
a particle (from Po) was known. The value of Q was determined
254 Looking In: Atomic and Nuclear Physics
Nuclear Reactions
255
from the observed increase in kinetic energy. The mass of tlic
neutron was thus found to ho 1,00(57 amu.
18.2 NUCLEAR FORCES; STABILITY OF NUCLEI
The hypothesis thai atomic nuclei sire composed of neutrons and
protons is now well established, and the term "nucleons" is used
to refer to these nuclear particles collectively. The size of the
nucleus is estimated by bombarding atoms with highenerLiy
electrons and counting Imw many of them score direct hits. The
radius of a nucleus containing .1 nucleons is found to be
approximately
Iin, n = 1.2 X ID »,!' m
(is. 2
An atom is stable because of the Coulomb force of attraction
which binds the electrons to the nucleus. Within the nucleus,
however, the Coulomb forces exerted by the protons are forces <>]'
repulsion which tend to make the nucleus unstable. The emission
of a particles from nuclei and nuclear fission (Chap. 20) are
evidence of this. Somehow the repulsive Coulomb forces within a
nucleus must be counterbalanced by strong attractive forces,
different from electrical and gravitational forces. The nature of
these nuclear forces is only partly understood. We shall discuss
some of the facts which arc known about nuclear forces.
An important, distinctive property of unclear forces is (heir
short range. The nuclear force between two nucleons becomes
negligible if they are separated by more than about 1,4 X 10 16 m.
In contrast, gravitational and electrical forces have no upper limit
on the distances over which they may act.
A second property of nucleus forces may lie deduced from a
graph of the binding energy per nucleoli /•;,, .1 against the number
of nucleons A (Fig. IS. I). Kxccpt for the lightest nuclei. E B .1 is
approximately constant, about 8 Mev per micleon. Thus the
total binding energy increases approximately in proportion to
the number of nucleons in the nucleus: K„ a A. (The relation
for a Coulomb force would be B« = A 2 .) This relation implies
that a given nucleoli is bound not to every other nucleoli present,
but only to its nearest neighbors. Then the addition of more
nucleons increases the total binding energy only by an amount
proportional to the number of nucleons added; K H /A does not
change appreciably.
Present evidence indicates that the nuclear force between two
protons is the same as the force between two neutrons and that
these may be equal to the force between neutron and proton.
The last property of nuclear forces which we shall mention is
pairing. The stable nuclei usually have even numbers of protons
10
■
* /
■ — f
20 40 60 80 1 00 1 20 1 40 1 60 T 80 200 220 240
A—
Fig. 18.1 Binding energy per nudeon as a function of mass number A.
and of neutrons (Table 18.1), Only the four light elements iH s ,
a Li°, sB 1 ", and 7 X U have odd numbers of both neutrons and
protons, and for these elements the numbers of neutrons and
protons are equal.
Table 18,1 Evidence
for pairing
Neutron
number (A — Z)
Proton number Z
Even Odd
Even
Odd
160 52
56 4
256 Looking In: Atomic and Nuclear Physics
When a plot of neutron number versus proton number is
made for all nuclei (Fig. 18.2), one observes a gradual increase m
the neutron/proton ratio with increasing Z. This is explained by
the fact that the Coulomb (repulsion) force between protons
increases more rapidly as the number of protons in the nucleus
increases than does the effect of the nuclear force between protons.
This difference in the behavior of the Coulomb pp force and the
nuclear pp force accounts for the gradual decrease in E B /A from
160
140
120
Nuclear Reactions
257
i
"* 100
■a
E 80
p
^ 60
40
20
t « —
/ •
/ •
/ s
/ s
/*
£ . ■_
Fig. 18.2 Neutron proton plot
for stoble nuclei.
20 40 60 80 100
Proton number, X
about 8.8 Me? for A near 50 to approximately 7.0 Mev for
A = 240 (Fig, 18.1).
18.3 NUCLEARREACTION EQUATIONS
We shall consider some possible outcomes when a particle or
nucleus ,r strikes a nucleus X resulting ha the emission of particle
?/ and the obtaining of nucleus J":
z + X— }■ +
(18.3)
The notation is often abbreviated as X(.c,y)Y, where the first
symbol stands for the struck nucleus, the symbols in parentheses
stand for the incoming and outgoing particles, respectively, and
the symbol following the parentheses represents the residual
nucleus. The reaction associated with Chadwick's discovery of
the neutron, Eq. (18.1), may thus be abbreviated as Be 9 (a,n)C™.
Before artificially accelerated particles became available, about
1932, only 10 nuclear reactions were known, all of the (a,p) type.
It seems probable that in the majority of artificially produced
unclear reactions the first step is the formation of a compound
nucleus. The projectile and the target nucleus coalesce. The com
pound nucleus is unstable, because of its excess energy. It emits
one or, sometimes, more particles of high energy to regain stabil
ity. When Rutherford bombarded nitrogen with a particles
emitted by Ra C (li)li)), he initiated the first nuclear transmuta
tion by artificial means. The equation describing it in terms of a
compound nucleus is
,He< + ,!?"♦ [«F l *J»«0" + ,11' (18.1)
The same compound nucleus (but not in flic same energy state)
could be produced by other reactions
The breakup of the unstable compound nucleus usually depends
only on its energy state, not directly upon the particle that pro
duced it. There are often several possibilities; for example,
U*Zn 6i l ^ w&l« + T
+, Cu" + ,H J
The 3 o*Zn Si may also eject other particles; ill 2 , iH*, sHe 1 or two
n', but the probabilities of these reactions are low. Present
nuclear theory does not permit prediction of the way a particular
compound nucleus will break up.
bike chemicalreaction equations, nuclearreaction equations
must be balanced. The total electric charge (the number of pro
tons) must be the same before and after the reaction. The total
number of nucleons (neutrons and protons) must be the same,
before and after the reaction. Together, these requirements mean
that the number of neutrons must be the same before and after
the reaction, likewise the number of protons. (There are two
258 Looking In: Atomic and Nuclear Physics
exceptions: If we regard ft* decay as a "reaction," then since there
is no incoming particle, the number of neutrons changes by ± 1
and the number of protons changes by + I. At extremely high
energies, greater than 2 Hcv, it becomes possible to create micleou
pairs. In such reactions, which we shall not discuss, the number
of nucleoli* docs not remain constant.)
18.4 THRESHOLD ENERGY
In a nuclear reaction .r 4 .Y — ► 1" 4 >j, the net increase in kinetic
energy is called the disintegration energy Q. This Q is the net
decrease in rest mass, expressed as its equivalent energy:
Q = H + m x )  (m v + Tn. u )]c l
Q = initial rest energy — final rest energy
(18.5)
Since Q is the amount of rest energy eon veiled into kinetic
energy, Q is often called the energy release of the nuclear reaction.
For an encounter which results in elastic scattering, Q = 0, If
the Q value of a reaction is positive, the reaction is called exo
thermic. Such a reaction can occur for incident particles of any
kinetic energy. If Q has a negative value, the reaction is called
emlolhermic.
Example. Calculate Hie Q value for the read ion T X 14 + n' » 7 ,V l * 4 y
,V  I i.odt.vji; 15.011)512
n' = 1.00898(5 X' 6 = i;).illMs7s
16.016512 m = O.OlKiSi amu
Q = 931(0.01 KM) Mev = H0.K Mev
Conservation of momentum imposes a condition on induced
nuclear reactions, as it does on all other collisions. This condition
is particularly important for reactions with negative Q value.
Prom energy considerations alone, one would think that ir the
incident particle x approached the target nucleus (at rest) with a
kinetic energy A'*.* = Q, then the reaction would occur. But then
the momentum would not be conserved. The initial momentum
is greater than zero, but the final kinetic energy, and thus the
final momentum, would be zero. So, actually the incident particle
must have enough kinetic energy B** so that the outgoing particles
can have the same total momentum as the incident particle. The
Nuclear Reactions
259
minimum value of A"*.., which makes the reaction possible is called
the threshold energy. The minimum value of /;**. z which satisfies
the equations for both conservation of energy and conservation of
momentum is found to be
Threshold  (£*.,) , ni n = (l + M Q (18.0)
Example. Find the threshold energy for the reaction
,jm + )H i _* ,()!* + p „i (q = _3.4 S m,. v )
Threshold = ( 1 + — ) Q = ( 1 + ^§1)3.48 Mev = 3.72 Mev
\ "».v/ \ 14.00//
PROBLEMS
1. State the number of protons and neutrons in each of the following
nuclei: ,[,i* 6 I!e"», fi C 13 , 1S S 3B , and n Ui tm .
2. The nuclear read ion
,l.i + ,II»>2.Hc* + Q
liberates 22.4 Mev. Calculate the mas, of JLa* in amu. (I)eutcron =
2.014180 amu, a particle = 4.00:3873 amu.)
3. Imagine that a free neutron gives off an electron and changes into
a proton. Calculate the energy Q which is consumed or liberated in this
process. What does your answer suggest about the stability of free
neutrons? •Ins, Q = 6.79 Mev
4. When neutrons :ire produced by bombarding deuterons with
dcutcrons, the reaction is represented by
,H*+ ,II i . ! lle a + o" 1 + Q
The neutrons produced in this reaction will have at least how uiurh
energy?
Am. 15 Mev plus the kinetic energy of the bombarding deuteron.
.">. As the source of I he energy radiated by stars, it has been sug
gested that a series of nuclear reactions such as this carbon cycle occurs:
C" + H' ..V + 7 Q =+l.»o Mev
X la » C 1 ' + e + + neutrino (K t $) , = 1.20 Mev
C" + H' — X 14 4 7 Q ' = +7.58 Mev
\u + H'^0" + 7 Q = +7.34 Mev
O lb * N" + e* + neutrino (A\ a),,,.,, = I .UN Mev
K' B + II 1 * C 12 + lb' Q = 44.98 Mev
Write the equation which represents the net result of this whole cycle.
Ans. 4H l — * He* 4 2e v + 2 neutrinos 4 energy
19
Absorption
of Radiation
Science has a social value, and the
man of science cannot wash his hands
of his discoveries. It is his duty to see
that they are used for the betterment
of mankind, and not for its destruc
tion, Q Fournier
To interpret experiments in nuclear physics and to apply the
knowledge gained from them, it is necessary to know how the
highenergy particles behave as they pass through mutter. For
this discussion, highenergy particle means one whose kinetic
energy is much greater than the ionization energy of the atoms or
molecules of the material in which it is passing. We shall discuss
the absorption of radiation chiefly in relation to the identification
of particles, the measurement of radiation dost;, and the prob
lems of human health.
19.1 TYPES OF RADIATION
Jn the behavior of a highenergy particle the most important fact
is whether or not it carries an electric charge. A particle which
260
Absorption of Radiation 261
carries an electric charge (as do the electron, positron, proton,
deuteron, and a particle) will exert a force on each electron near
which it passes. A charged particle collides with many electrons
in traveling even a short distance in matter. In many of those
collisions, the struck electron is knocked out of its atom. The
incident charged particle loses its kinetic energy as it leaves
behind a trail of ion pairs (ejected electron and ionized atom). A
stream of charged particles is referred to as an ionizing radiation.
Photons and neutrons which carry no charge do not necessarily
collide with every electron near their paths. Streams of uncharged
particles are called nonionizing radiation.
19.2 DETECTORS
Ionizing particles are easy to detect electrically. In an ionization
chamber, a metal cylinder C has a wire II' insulated from the
cylinder along the axis. The tube is filled with gas at low pressure,
and a potential slightly less than that reiumed for a discharge is
maintained between cylinder and wire. A thin window allows
particles, say, a particles, to enter the chamber. Kach particle
ionizes the gas, producing a rush of charge and a fall of potential
at P which actuates a counter circuit. Thus one can count the
number of a particles. The behavior of the ion pairs created can
be studied by plotting a curve of the size of the current pulse
versus the voltage applied to the tube. The ionization chamber
Bottery ;
1
Capacitor
#
To amplifier
and counter
resistor
Fig. 19,1 Ionization chamber particle counter.
(Fig. 19.1), the proportional counter, and the Geiger.Muller
counter are ionization instruments designed to operate on dif
ferent regions of the curve.
A scintillation counter makes use of one of several substances
262 Looking In: Atomic and Nuclear Physics
which, when struck by a single particle, convert some of the
energy received in the collision into visible tight. About. ]«)()•(
investigators <rf radioactivity watched and counted the flashes of
light which individual a particles produced in zinc sulfide. Since
l!)l I a scintillator or phosphor such as a clear crystal of naph
Al foil reflectors
Photo cathode
semi transparent
First dynode
Tenth dynode JJ "£ "..j—  " Col lector grid
Output
Fig. 19.2 Scintillation counter.
thalene has been used in conjunction with a pilot onmlfiplier, for
automatic counting. A particle or a yray photon entering the
phosphor causes a flash of light which is reflected by the aluminum
foil onto the photocallmde. Klcctrons are emitted from it, and
these are subsequently multiplied to produce a relatively large
pulse at the output of the tube.
A cloud chamber, invented by C. T. It. Wilson in 18117, per
mits us to see the path of a particle through a gas. It consists of
an enclosure filled with air and some vapor at a temperature
just above, the condensing temperature. The chamber is designed
so that its volume may be suddenly increased. This expansion
Absorption of Radiation 263
depresses the temperature of the vapor below its "dew point."
Some of the vapor will now condense. A vapor condenses prefer
entially on charged particles, as nuclei for droplets, if there are
any present. So, if the gas has been traversed by a particle which
ionized molecules along its path, the vapor will condense on these
ions and the path of the particle will be visible as a trail of liquid
droplets.
Photographic plates were used by Bccquerel in his discovery
of radioactivity (1886). Recently the manufacture of special
emulsions for nuclear research has revived the use of this type of
detector. Nuclear emulsions contain about 10 times the concen
tration of silver halide as do ordinary photographic emulsions,
and are much thicker. Xuclear emulsions can be made sensitive
to slow neutrons by incorporating small amounts (I per cent) of
lithium or boron, which undergo an (n,a) reaction. Emulsions
may be "loaded" with other elements (such as uranium) to study
specific reactions. In film badges, the general darkening of the
photographic emulsion, on development, measures cumulative
exposure to radiation. In autoradiography, the distribution of
radioactive material in a tissue or mineral section is determined
by placing the specimen in contact with a photographic plate, in
the dark, and developing the resulting pattern.
The bubble chamber, invented by D. A. Glaser in 1952, takes
advantage of the instability of superheated liquids for bubble
formation, much as the Wilson cloud chamber uses the instability
of supercooled vapors for droplet formation. The cloud chamber
and the bubble chamber have similar general characteristics as
particle detectors. The resetting time is longer than lor counters.
The advantages of the bubble chamber lie in the high density
(greater absorption) of its sensitive material and its ability to
recycle in a few seconds. Bubble chambers filled with liquid
hydrogen offer simplicity in interpreting collisions with protons,
without contaminating elements.
19.3 DETECTION OF NEUTRONS
A neutron is attracted to other near nucleous by the nuclear
force, but it is neither attracted nor repelled by an electric charge.
Since a neutron and an electron exert no forces on each other,
they do not collide. (We can neglect for practical reasons the
264 Looking In: Atomic and Nuclear Physics
extremely small gravitational force between an electron and a
neutron and also a small electromagnetic force associated with
the magnetic moments of the two particles.) Since nuclei occupy
only a small fraction of the volume of matter, neutrons are pene
trating radiation, traveling relatively large distances between
collisions. When a collision does take place, either the neutron
is scattered or a nuclear reaction occurs.
Since neutrons do not betray their presence directly in de
tectors (Sec. 19.2), they must he detected by the ionization which
results from some nuclear reaction of scattering. For slow neu
trons (having kinetic energy less than I ev) it is convenient to
use the reaction
JB»f oNi» Ji» ,He l
If a counter tube is filled with a gas containing boron, BK 3 , or if
the wall is coated with boron, then some neutrons will he captured
to give fast a particles, which will cause ionizations in the gas
and give counts.
Another method used to detect slow neutrons makes use of
the reaction
on 1 + ^In' 16 * win" 6 + y
The radioactivity of an indium foil after exposure to a neutron
beam is a measure of the number of neutrons which passed
through the foil. The (n,y) cross section, or probability of cap
ture, is sharply higher for neutrons of l.4(Sev energy. Thus this
detector favors or picks out those neutrons.
The detection of fast neutrons, and the initiation of certain
important reactions, often requires first that the neutrons be
slowed down. This is accomplished by arranging for the neutrons
to pass into a moderator— a material such as graphite or D,,0 in
which the probability (cross section) for scattering is much larger
than that for a nuclear reaction. The neutrons then bounce
around among the nuclei until both reach an average energy of
!i/,r, where k is the Boltzmaun constant.
Bxampk. Find the energy of a "thermal neutron" in n moderator at
22 C
B k = (1.3H X 10" joulc/K°)(295°K) • 6.11 X 10~" joule
= 0,0382 ev
Absorption of Radiation 265
19.4 ABSORPTION OF PHOTONS
Photons can interact directly with the electrons of the material
through which they pass. But for highenergy photons, the cross
section (probability) of such interact ions is so small that the
photons constitute an extremely penetrating radiation. The
energy of photons can lie dissipated in three different kinds of
collision.
In the photoelectric effect a photon is absorbed by an atom; its
energy is used to eject an electron and to impart kinetic energy
to the electron. The cross section for the photoelectric effect
increases rapidly with increasing atomic number (Z) and de
creases rapidly with increasing energy (hr) of 6he photon.
In pair production, the energy of the photon is converted into
a positron and an electron and their kinetic energies. The cross
section for pair production increases rapidly with increasing Z
of the absorber and with increasing energy of the photon, above
the threshold value of 1 Mev (= 2m c 2 ).
In the Campion effect, photons are in effect scattered, not
absorbed. A photon is still in play after the collision. The cross
section is a slowly varying function of (hv) and Z.
The detection of photons is relatively simple; for any type of
collision described above gives a fast electron: a photoelectron,
a Compton electron, or an electronpositron pair. The electrons
are ionizing particles and may be counted directly.
The variation of photon "absorption" by each of these proc
esses is represented in Fig. U)M, where for each process, an absorp
tion coefficient a is defined as the product of the cross section a of
the reaction and the number n of atoms per unit, volume, a — rur.
If the Compton effect were strictly an absorption, a total absorp
tion coefficient a, could be defined for photon absorption
Ctt — OfphutM "T" G^air I ^Complin
ami the attenuation of a beam of x rays or 7 rays could be repre
sented by the exponents! equation
While this relation has practical usefulness, it must be applied
with care, since eeoinpton does not relate to a true absorption.
266 Looking In: Atomic and Nuclear Physics
a, cm" '
.
1.4
i I l a niol
\ ^*m
^ 1 J*
1.2
■ ^ 1 1 /?
\ ' \ A
1.0
0.8
\ 1 V Jya. ra „
» x \ A
0.6
 \%~4
0.4
V s /
0,2
v^ »»_
0.5 5 50
Photon energy
500 Mev
Fig. 19,3 Variation of photon absorption coefficient, a, in
load, with photon energy.
19.5 RANGES OF HEAVY CHARGED PARTICLES
Charged particles heavier than electrons experience frequent,
collisions with electrons in passing through matter. The heavier
particle cannot lie appreciably deflected, and il can lose only a
small fraction of its energy in collision with an electron. Vet the
collisions are so frequent that charged particles are slowed down
to thermal energies in very short distances. Charged particles are
not a penetrating radiation. A proton with 10 Mev of kinetic
Distance traveled, S
Fig. 19.4 Kinetic energy vs. distance troveled for a charged
particle.
Absorption of Radiation 267
energy travels only 0.0 mm in aluminum; a 10 Mev a particle
travels only 0.00(> mm in aluminum.
The decrease in the kinetic energy of a charged parlislc with
distance traveled is indicated schematically in Fig. 19.4 as
occurring in many small steps. The distance traveled before the
kinetic energy is all lost is called the range of the particle. Range
depends on the particle, its initial energy, and the absorbing
material. When the kinetic energy of the charged particle has
been reduced to a small value (about 100 ev for a proton), it
becomes increasingly probable that the ion will capture an elec
tron and end as a neutral atom.
19.6 ABSORPTION OF ELECTRONS AND POSITRONS
The path of an electron or positron is longer than that of a heavy
charged particle of the same energy, but it is a path full of bends
because of scattering. Electrons, like other charged particles, lose
their energy in a very small region of space; they do not constitute
a penetrating radiation.
19.7 RADIATION DOSE
The dose of any kind of radiation received by an object is the
amount of energy that the object absorbs from the radiation.
One might try to use a calorimeter to measure the energy ab
sorbed by a specimen in terms of the resulting rise in its tempera
ture. It turns out that even a lethal dose of radiation produces an
undetectable rise in the temperature of a biological specimen.
Radiation produces many specific effects on physical, chem
ical, and biological systems. Many of these effects seem closely
related to the ability of the ionization caused by the radiation to
promote particular chemical reactions. Hence methods have been
devised to specify dose in terms or ionization.
A beam of x rays or y rays is said to give a dose of one roentgen
(1 r)* if it will cause 2.08:{ X 10" J ionizations in 1 cm 3 of dry air at
"The National Bureau of Standards Handbook H47 gives the defi
nition: "The roentgen shall be the quantity of x or y radiation such that
i he associated corpuscular emission per 0,001293 gin of air produces, in
air, ions carrying 1 esu (if quantity of either sign," The figure 0.001293
268 Looking In: Atomic and Nuclear Physics
Absorption of Radiation 269
0°C and 1 atm. An ionization chamber is used to measure the
dose from the radiation. The radiation passes through the air
between the plates, and the ionization occurring in the air is
collected. The chamber and its electrometer can be calibrated to
read directly in roentgens. A widely accepted human tolerance
dose rate is 0.3 r per week. The dose from cosmic rays at the
surface of the earth is about 2 per cent of this tolerance dose.
The roentgen was defined for photons. To extend the unit to
permit measurement of radiation dose from other particles, and
in living tissue, the roentgen equivalent physical (rep) is desig
nated as the radiation which produces the same energy as one
roentgen of x or 7radiation. This amounts to 97 ergs per gram
of tissue. This value is based on the observation that for any
particle and any gas the average energy lost by a fast charged
particle per ion pair formed is about 33.5 ev, A third unit for
radiation dose is the red: the radiation which produces LOO ergs
per gram of tissue.
19.8 BIOLOGICAL EFFECTS OF RADIATION
Living tissue is damaged 1 >.V exposure to highenergy radiation.
The danger is insidious, for the observed biological effects may
be delayed for periods ranging from a few days to years, depend
ing upon the type of radiation and the dose received. Among the
effects of overexposure to radiation are a decrease in the number
of white blood cells, loss of hair, sterility, cancer, cataracts
(chiefly from neutrons), and destruction of bones. In addition to
the damage to the person receiving the radiation, there may be
genetic effects extending through many generations of offspring.
Penetrating radiations are effective in producing mutations or
changes in heredity.
X rays, y rays, and particles from supervoltage accelerators
penetrate tissue readily and constitute externa! radiation hazards.
In general, « and /S particles have low penetrating power, and
Kin is the muss of I cm 3 of dry air at 0°C and 1 atm. Since 3 X 10 s esu
of charge = 1 coul, I r produces
1
3 X 10» statcoul/coul 1.6 X 10" coul/ion
= 2.083 X 10» ion pairs/cm'
damage from external sources will be confined to a thin layer of
tissue. But a and emitters become internal hazards when intro
duced into the body in foods or otherwise.
The various kinds of radiation damage seem to he statistical
in nature, with no threshold or "safe" minimum exposure below
which no injury occurs. Hence it seems prudent, to avoid all
unnecessary radiation exposure. Since some exposure may be
necessary for some people, responsible agencies have suggested
tolerances, such as a wholebody exposure of 0.3 r per week when
continued over a long time. I'or hands and feet the tolerance may
be 1.0 r/ week. A single exposure of 25 1 in an accident can prob
ably he accepted. A wholebody exposure of about 500 r would
probably be fatal, statistically, to 50 per cent of persons so
exposed.
19.9 ATMOSPHERIC CONTAMINATION FROM NUCLEAR
WEAPONS TESTS
The probable effects on the health of the world population of
atmospheric contamination arising from nuclear weapons tests
cannol be assessed reliably from data known at present. Vet on
the basis of incomplete information and conflicting interests,
political decisions about nuclear detonations must be made which
vitally affect our national defense and the freedom and health of
generations to come.
If one examines, in addition to research reports, some 10
official statements made since 1 !>.">(> by the Congressional Joint
Committee on Atomic Energy, the United Nations Scientific
Committee on the Effects of Atomic Radiation, The National
Research Council, and the (British) Medical Research Council,
one finds that these responsible bodies are in agreement on the
following points:
1. Radiation exposure of the world population from fallout
(including Si'"") as a result of tests through mid1963 is small
compared to natural background radiation and other man
made radiation (such as diagnostic x rays).
2. Any amount of radiation, however small, may carry a small
but finite risk of increasing the genetic mutation rate of the
population.
270 Looking In: Atomic and Nuclear Physics
3. Tt is unknown whether or not there exists! a threshold radiation
dose for the production of somatic effects, including leukemia,
bone cancer, and general life shortening.
4. Calculations of biospheric contamination in the event of con
tinued testing of nuclear weapons are intelligent guesses at
best, since conclusions depend on the many assumptions that
must be made.
5. Continued testing of nuclear weapons will increase biospheric
contamination and consequent risk to the world population.
Accelerated testing as more nations become nuclear powers,
and (he touching off of nuclear war, could result in a serious
radiation hazard to world health.
19.10 DISPOSAL OF NUCLEAR WASTES
Nuclear power ranuol be developed by present techniques with
out also producing radioactive waste materials which are harmful
to man. The safe disposal of such radioactive wastes is far more
difficult than that of ordinary industrial wastes. More than
ti;5 million gal of highly radioactive nuclear wastes are now con
fined in milliongallon underground tanks because they are too
"hot" to dump. Although the concrete and steel tanks are ex
pected to last several decades, their contents will still be too
radioactive to dump when the (auks have deteriorated!
There has been increasing local public protest against the
dumping of nuclear wastes into the oceans, relatively close to the
shore ; particularly by citizens ul Cape Cod, Texas, and Mexico.
It has also been pointed out that it may even be dangerous to
dump nuclear wastes in remote and deep trenches of the oceans
because (I) experiments increasingly indicate thai there is con
siderable circulation of ocean waters and (2) marine organisms
tend to build up small and nearly harmless radioactive levels in
sea water to potentially dangerous levels in the food supply.
At the present time there are four general sources of radiation
which can harm the present and future generations. In order of
intensity, these are (1) medical and dental x rays, (2) radioactive
sources naturally present in the earth, (3) radioactive fallout from
nuclear testing, and (4) waste products from nuclear reactors.
Within a decade or two, the latter two sources of radiation
exposure may become the most important.
Absorption of Radiation 271
SUGGESTED READING
Articles in the Bulletin of the Atomic Scientists.
The Milk We Drink, Consumer Reports, March, 1959.
Fallout, in Our Milk, Consumer Reports, February, 1960.
The Huge and Kveriiiereushig Problem of Radioactive Wastes, Con
sumer Reports. February, I !)(!().
Fallout 1963 . . . an interim report, Consumer Reports, September,
19(iH.
I^utgham, Wright, and B. 0. Anderson; "Biospheric Contamination
from Nuclear Weapons Tests through 1968," Los Alamos Scientific
Laboratory, University of California, I.os Alamos, X.Mex. 100 pp.
Contain.* bibliography of 7 I ilems.
I often say that when you can measure what you are speaking about and
express it in numbers, you know something about it; but when you cannot
express It in numbers, your knowledge is of a meagre and unsatisfactory
kind; it may be the beginning of knowledge, but you have scarcely, in your
thoughts, advanced to the stage of science, whatever the matter may be.
Lord Kelvin
Life would be stunted and narrow if we could feel no significance in the
world around us beyond that which can be weighed and measured with the
tools of the physicist or described by the metrical symbols of the mathe
matician. Sir Arthur Eddington
Accurate and minute measurement seems to the non scientific imagination
a less lofty and dignified work than looking for something new. But nearly
all the grandest discoveries of science have been but the rewards of accurate
measurement and patient longcontinued labor in the minute sifting of
numerical results. Lord Kelvin
It does not take an idea so long to become "classical" in physics as it does
in the arts. K. K. Darrow
20
Unconventional
Energy Sources
. . . the discovery with which we are
dealing involves forces of a nature too
dangerous to fit into any of our usual
concepts. Congressional Record,
T87S, commenting on the gasoline
engine
A physicist, like other persons, often finds living more purposeful
and satisfying when he haw both short and longrange goals.
Some physicists seek to relate their goals to some of civilization's
longrange problems: food production, world peace, education,
and the exploitation of new sources of energy. It would seem that
physics could contribute most directly in finding new sources of
energy to supplant depleted reserves of coal and oil and to meet
the ever increasing demand for power for industry, transporta
tion, and the home. Since we never create energy, it might be
more precise to speak of a search for new and practical energy
conversion devices.
Some possible sources of energy are so speculative that they
are referred to as esoteric sources. The term "unconventional" is
272
Unconventional Energy Sources 273
reserved for those untapped sources ahout which enough is
understood today so that one may reasonably predict that
engineering refinements will soon make of them practical energy
sources, important in our economy. Nuclear reactors, thermo
electric, thermionic, ami magnetobydrodynamic generators, solar
cells, and fuel cells give promise of becoming increasingly impor
tant practical sources of energy.
20.1 NUCLEAR FISSION
When, in 1042, the book "Applied Nuclear Physics" (K. Pollard
and W. L, Davidson) was published, its title sounded visionary.
Since then we have witnessed important and varied applications
of nuclear physics. The nuclear reactor has heen developed into a
practical source of electric power. (A reactor may become the
ultimate source of power for space travel.) With particle acceler
ators and nuclear reactors, a host of new isotopes have been
created. These have been important in further fundamental
studies. They have also found diverse practical applications.
In 1934, Fermi and his collaborators attempted to produce
elements beyond the normal limit at uranium. In bombardment
of the lighter elements by slow neutrons, the element after the
capture is usually transformed by electron emission into the ele
ment of next higher atomic number. Therefore, one might expect
that a similar bombardment of uranium (Z = 92) would produce
a new element (93). This reaction has been produced with
neptunium (93) as the resulting product. Neptunium also dis
integrates by emitting a (i particle to produce plutonium (94).
Plutonium is a rather stable clement having a halflife of 24,400
years. From 1944 to 1950, four other new elements were produced
in the cyclotron: americium (95), curium (96), berkclium (97),
and californium (98). More recently elements einsteinium (99),
fermium (100), mcndelevium (101), and nobelium (102) have
been reported.
In 1939, Halm and Strassmann found one of the products of
neutron bombardment of uranium to be a radioactive barium
sijBa 139 . There must then be another fragment such as 36 Kr associ
ated the barium fragment to make the charges equal. Ncir
separated the isotopes of uranium in a mass spectrograph and
found that «U s,b is the one that undergoes the splitting process
274 Looking In: Atomic and Nuclear Physics
called fission. Fission is a new type of radioactive process, the
first that produced particles more massive than a particles.
In the process of fission of uranium there is a decrease in total
mass, and therefore there is a corresponding gain in energy. Such
a reaction then is a possible source of energy. This energy is con
trollable since the process can be started at will and its rate can
be governed.
Among the products of fission one finds one to three neutrons.
These neutrons are faster than the ones used to start the fission,
but if they strike uranium nuclei, they can cause fission. Since
the fission produces the starting particles and releases energy,
the reaction can perpetuate itself, provided there is enough
uranium present so that the neutrons produced will hit other
uranium nuclei. Thus a chain reaction can be set up. The smallest
amount of material in which a chain reaction (constant neutron
flux) can be set up is called the critical mass.
20.2 NUCLEAR REACTOR
A nuclear reactor is a device for utilizing a chain reaction for
any of several purposes : to produce power, to supply neutrons,
to induce nuclear reactions, to prepare isotopes, or to make
fissionable material from certain "fertile" materials. Typical
components of a reactor are: the fissionable fuel (LI or Pu), the
moderator (graphite or D a O to slow down the fissionproducing
neutrons), the control rods (usually Cd strips, whose insertion
captures neutrons and slows the fission rate), and the coolant
(water, air, hydrogen, or liquid metal, such as \a).
In power reactors, the coolant, through a heat exchanger,
may furnish steam to operate a conventional turbine and elec
trical generator. Breeder reactors make new nuclear fuel from
fertile substances which cannot themselves sustain a chain reac
tion but which can be converted into fissionable material. One
possible breeding reaction is
iNP^^MPO** (201)
T ,
* t^l.
2'A ruin
'2:.i duy»
20.3 FUSION
Nuclear energy can also lie released by fusion of small nuclei into
larger nuclei if in this process there is a decrease in mass. In such
Unconventional Energy Sources 275
a process the two positively charged nuclei must come into con
tact even though there are strong electrical forces of repulsion.
This requires thai I lie particles he moving with high speeds. With
artificial accelerating apparatus, a few nuclei are given very high
speeds. Only occasionally will such a particle strike another
nucleus before it has lost too much of its energy to make contact.
Thus the process is extremely inefficient, and more energy must
be supplied to initiate the fusion process than is realized from the
reaction.
The necessary condition for a controlled nuclearfusion process
is the attainment of high particle energies for a time interval long
enough to bring about kinetic equilibrium. Knergy must be sup
plied initially to attain temperatures about 2 X 10 7 °K (at which
thermal fusion occurs in stars). At the same time reactants must
be confined. Ordinary walls will not suffice, for they would
vaporize under bombardment of highenergy particles, and these
would be quickly cooled below their fusion temperature. These
problems of heating and confinement must be solved in any con
trolledfusion reactor.
The choice of fuel for a eon t rolledfusion reactor is made on
the basis of availability and the probability of attaining with it
the necessary high temperature. One would prefer elements of
low atomic number because of the low Coulomb barrier to be
overcome in the fusion reaction. Possible fusion reactions are
shown in Fig. 20.1.
Initial heating first strips the electrons from the atoms to pro
duce a "fourth state of matter," a fully ionized gas, or plasma.
Further heating of the plasma is done by adding electric energy,
in part by using the resistance of the plasma to produce familiar
Ohmic (or Joule) heating.
Suitably designed magnetic fields provide a sort of magnetic
bottle to confine the ions at I0*°K. In the pinch effect, a cylin
drical current (10° amp) contracts because of electro magnet it
forces (parallel currents attract each other). The plasma inside is
thus compressed, producing very high temperatures. The simple
pinch is unstable, but with suitable stabilizing fields thermo
nuclear temperatures have been attained for confinement time
of about 0.001 sec. Thus far, however, the power required for
these devices has exceeded the useful power gained from the
fusion process.
THE FOUR STATES OF MATTER
1Solid
2 Liquid
3 Gas
4Plosrao
tlili
First .hres itafei of roaMer vory with arrongenien.
and movement of maleculei, ihe ima'lleir parr.c1.ev
C K« roc t •>■ lii ; c of a rrva i«f io I . tn to I id , mo lee u 1 ei
or* cfoYepackec 1 ond trill ■ In liquid they tnovt
about within limiti. In a got, moleculei ore man
(coflered Qnd movv Foiier, Fooith lfoie^ ploinna, it
wholly '"ionized" 901. Molecule* break into aTorm,
alonm into poiilive ion* and regaive elecrron>>.
• Proton
O Neutron
THE FUSION REACTION
Deuteriurn Fusion
He3 +
P
Energy
En ergy
3,25
Mev
■ 4 Mev
Pi^rC^
Deuterium tritium fusion
He*
^^5 M^
fusion con take piece within a plasma, Fuiian ii
combination of nuclei (atom* minus electrons} of
certain lighl element!. Man of the Fuiion product!
rl let* than that of orioinal nuclei; the difference
h radiated os energy, mojlly heat. The fcjn"« voit
1 HEATING THE PLASMA
■;'..:  Direction of current.
. •■■....
To get controlled ihermonucleai reaction in
stead of explosion, small quonliriei of plasma
must be contained and heated. Process begins
with passage 0! a current through the plasma
inside a Straight nr doughnut shaped tube.
energy is from hydrogen fusion . On earth
most likely such reaction involves Twtopes
(voiiants) of hydrogen — deuterium and
tritium. In o plasma heated to millions of
degrees, they may fuse, as in the Hbomb.
THE PINCH EFFECT
PROBLEM OF INSTABILITY
mm
A tuirenl eieaies o magnetic field around itself.
This Field exerts pressure on plasma, "pinching,"
it toward center,, compressing St, making it hotter
and preventing plasma pari ides from touching
walls af tube . But this is theoretical behavior.
CONTROLLING INSTABILITY
In practice pinched column of ploimo develops
"kinky," Pinch wandert llighHy; distortion af
■he magneric Held create* new forcer and diirorn
column further. Pinch eilher rouchei walk of r^re
rub* and loiei energy (A) or ii broken or or (B) .
Powige of new current (A) around rube create! a
linear meaner ic field in column/ giving il "back
bane/' Currenli induced in wo 111 of lube (B) help
uraighren column. Pinch can then be mointoined
longer.
Fig. 20.1 Principles of a thermonuclear reaction, f Copyright by Trie New/ York
Times. Reproduced with permission.)
276
Unconventional Energy Sources 277
20.4 THERMOELECTRIC CONVERSION
The direct conversion of heat to electricity on a commercial scale
is a prospect that has fascinated scientists and engineers for
decades. In 1821, Thomas Sccbeck noted thai heal applied to
one junction of a circuit containing dissimilar metals would cause
a small electric current in the connected circuit. The physical
median ism can be understood, qualitatively, in terms of the free
electron picture of conduction. Kach metal contains some free
electrons. These electrons can be made to move by an electric
field or by a thermal field. If heat is applied at one end of the
conductor, the electrons will rearrange to become somewhat more
sparse in the warmer regions of the .specimen and more dense in
the colder regions. This leads to an electrical gradient. To take
advantage of it, the circuit is closed through a dissimilar metal
(Fig. 20.2). Then, as long as the temperature difference is main
tained, the difference in electrical gradient in the two conductors
will cause an electron flow, here clockwise.
The efficiency of conversion, using the best metal combina
tions, was only 1 to 3 per cent. Thus, until recently the only
practical application of Seebeck's effect was in thermocouples to
measure temperatures. Recent discoveries in the field of semi
conductors have led to substantial improvement in thermoelectric
conversion efficiency and foreshadow practical thermoelectric
generators of power. One arm of the thermocouple may be made
of an «type semiconductor, in which the voltage difference is
established by the flow of negatively charged electrons. The other
arm may be a ptype semiconductor in which the voltage differ
ence occurs by the flow of positively charged voids (holes)
vacated by the electrons.
The attractiveness of materials for thermoelectric converters
can be specified by a figure of merit Z defined as
Z = 4
(20.2)
where T = temperature, °K
S = Seebeck coefficient, volt/K° (i.e., emf developed per
unit temperature difference in the specimen)
r/ = electrical conductivity, (ohmcm) 1 (i.e., reciprocal
of resistivity p)
k = thermal conductivity, watts/ C° cm
278 Looking In: Atomic and Nuclear Physics
Both 8 and a depend on the density of conduction electrons in
the specimen, as shown in Fig. 20M. It is apparent that for
intermediate and low temperature use, semiconductors will pro
vide the highest efficiency in thermoelectric converters. Pairs of
semiconducting comp< ds which have high conversion effi
I mutators
Semiconductors
Metals
Fig. 20.2 A thermocouple circuit of dissimilar metals, A
and 8. The migration of electrons from regions of higher
density toward regions of lower density produces o con
ventional current in the counterclockwise sense.
ciencies have been found by making binary ami ternary com
pounds of materials in groups I, III, and VIj or the periodic
table: AgSbSe^, CuTiSt, etc. The numerous combinations possible
make the task of screening and developing the most favorable
thermoelectric materials a formidable one. Vet exciting progress
10"
Electron density, no./cm*
Fig. 20.3 Properties that govern the choice of materials for thermoelec
tric devices, (Courtesy John C. Kelly, VVesfinghouse Research laboratories.)
40
 30
c
HI
9
a.
£. 20
o
c
.2
'o
LU
10
Practical limits of thermoelectricity
Central station
esel or marine
Present ^^ ^S Automobile
Auxiliary power
J !
L.
L.
10 100 1000 10,000
Power rating, kilowatts
100,000
Fig. 20.4 Thermoelectric power devices con be competitive with other power
sources. {Courtesy John C. Kelty, Weslinghouse Research laboratories.)
279
Fig. 20.5 Power producing thermoelectric elements mode of germonium
silicon semiconductors. (RCA laboratories, Princeton, N.J,)
Heat source
Nuclear
Nuclear heat transFer
Fossil Fuel
^"Waste" heat
r
Junction technology
Thermoelectric
materials
Controls
DCmotching voltage/
current inverters
Fig. 20.6 Thermoelectric power system alternatives. {Adapted from the Genera/
Electric brochure GEZ3Q79B.}
280
Unconventional Energy Sources 281
has been made. Seebeck's original thermocouples (1821) could
convert heat into electric power with an efficiency of only 2 per
cent. Study of the PbSZnSb couple by Maria Telkes in 1833
raised the efficiency to 4 per cent. Further work with semi
conductors has given the present efficiency of about 17 per cent.
Theoretical considerations (using quantum mechanics) suggest
that it will be possible to attain efficiencies as high as 35 per cent.
This will make thermoelectric power devices competitive with
existing power sources (fig. 20.4). Each of the germaniumsilicon
thermoelectric elements shown in Fig. 20.5 is capable of gener
ating about i watts upon exposure to heat at about 1000°C. A
s<]uarefoot platelike arrangement of snch elements could generate
up to 10 kilowatts, nearly three times the usual electric power
demand in a home. Some alternatives to be explored in the
development of a thermoelectric power system are suggested in
Fig. 20.0.
20.5 THERMIONIC CONVERTER
Thermionic emission was noticed by Edison in 1883. In 1956
V. C. Wilson designed a converter in which electrons are "boiled
J^Xv. Cooling
Insulator
1 "* ia — 1 —
Anode
f Electrons 
MM*
Cathode
Load
KS Heat
Fig. 20,7 A thermionic converter.
out" of a hot metal and used to produce an electric current
directly. One obvious difference between the thermionic con
verter (Fig. 20.7) and the thermocouple is that in Wilson's
device the metals arc separated by a vacuum or a gas at low
pressure. There is electrical Row between the electrodes, but there
282 Looking In: Atomic and Nuclear Physics
is less flow of heat in this space than through a metal. Thus t ho
electrodes can be at different temperatures, and the efficiency is
increased.
The conversion process is shown in l'"ig. 20.8, where electron
energy is plotted against distance from cathode to anode. The
base line corresponds to the energy of the electrons in the cathode.
Heating the cathode "lifts" some of these electrons over the work
function barrier at. the surface of the cathode, w n into the space
between electrodes. If the electrons can follow path a to the
anode with only a small loss of energy, there will be a potential
rs
I V
I \
I \ b
I V
\
~~ «^  <
Cathode
> 140Q°K
Fermi level
77777777777777^
.1
a
gas
^ Plasma drop
 T Anode ~ 700°
J" Fermi level
' Load
— r
Output
voltage
Fig. 20.8 A plot of electron energy vs. distance (cothode
to anode) in a thermionic converter.
difference between the electrodes, capable of doing work in an
external circuit. In vacuum devices, the electrons entering the
interelectrode space soon form a spacecharge barrier, represented
by path h. This would increase the cathode electron energy neces
sary to electrons to cross to the anode, so the space charge is
neutralized by adding an ionizable gas, such as cesium. Or
alternatively a vacuumtype converter is made with a very small
{0.001 in.) spacing between cathode and anode to minimize space
charge effects.
Current models of thermionic converters are stated by Gear
eral Electric to have these characteristics: vacuum type, efficiency
5 per cent, cathode temperature 1100°C; gasfilled type, effi
ciency 17 per cent, cathode temperature 15:!0 °C. The gaslillcd
Unconventional Energy Sources 283
unit has the additional advantage of smaller weight per unit of
power: ! versus 2."> li> kilowatt.
20.6 MAGNETOHYDRODYNAMICS
An Mill.) generator utilizes the principle discovered by l'araday
that an ion moving in a magnetic held experiences a side push
(Sees. 10.8 and 10.9), Hot ionized gas is forced between the poles
of an electromagnet (Fig. 20.9), producing a voltage difference
To regenerator
Hot gos Flow
Flow
©
Field
Current
JV
VW
Fig, 20.9 A magnetohydrodynamic generator,
between the electrodes, at right angles to the magnet. By con
necting the elect rudes, power may be delivered to an external
load. A regenerator is used to recover energy from the emerging
Kas stream which may still be as hot as 2000°C when its ionization
has dropped to levels insufficient for effective energy conversion.
An MUD generator might be operated as part of a conven
tional gas or combined gas and steam turbine cycle. Few data
exist today on which to calculate efficiencies attainable with such
a combination . Some estimates suggest that addit ion of an M H D
284
Looking In: Atomic and Nuclear Physics
generator could raise tlio overall efficiency of a generating station
to 55 per cent.
20.7 FUEL CELLS
A fuel cell is a continuousfeed electrochemical device in which
the chemical energy of reaction of a fuel and air (oxygen) is con
verted directly and usefully into electrical energy. A fuel cell
differs from a battery in that (1) its electrolyte remains un
changed and (2) it can operate continuously as long as an externa!
supply of fuel and air is available.
Sir William CSrove, an Englishman distinguished in electro
chemistry and the law, used a hydrogen fuel cell in his experi
Chemieol
energy
r
* Heat
\
*■ Thermoelectric
*■ Thermionic *
*■ Thermogolvanic
Fuel cell
/
Fig. 20.10 Fuel cells convert chemical energy directly into electric energy,
thereby avoiding the thermodynamic limitation on the efficiency of heot engines.
ments as early as 1839. By the end of the last century, Wilhelm
Ostwald and others came to appreciate, through thermodynamic
analysis, that the fuel cell is potentially the most efficient simple
way of converting chemical energy into electrical energy.
Heat engines are subject to the Carnot limitation of thermo
dynamics which says that the maximum theoretical efficiency
with which heat can be converted into another form of energy is
determined by the inlet and exhaust temperatures of the engine:
7> 71
J i nlet / outlet , on rt\
Maximum efficiency =
7',,,
i,i
It is an attractive feature of the fuel cell that its efficiency is not
subject to the Carnot limitation, for the energy being converted
never deteriorates into the random motion of heat. The fuel cell,
when compared with familiar methods of generating electric
1
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285
286 Looking In : Atomic and Nuclear Physics
Fig. 20.12 A 75wott 4cell AllisChalmers fuel cell system designed
for and tested under "jsero gravity" conditions.
energy (Fig. 20.10), is very direct in its conversion of chem
ical energy into electrical energy. Partly because of this incentive,
fuel cells are probably the most highly developed of the uncon
ventional energyconversion methods discussed in this chapter.
Under favorable conditions, efficiencies of 80 and even 5)0 per cent
have been reported with hydrogen fuel.
A fuel cell, like any other electrochemical cell, contains two
electrodes: anode and cathode. These are joined externally by a
metallic circuit, through which the valence electrons from the
fuel flow, and internally by an electrolyte, through which ions
flow to complete the circuit (Fig. 20.1 1). These are the electrode
reactions :
Anode 2H 2  411+ = ee~
Cathode Oj 4 4H++ ier = 2H 4
Overall O* + 211, = 2H a O
Unconventional Energy Sources 287
The electron does useful work for its in passing from anode to
cathode in the external circuit. The hydrogen ion completes the
circuit by going from anode to cathode through the electrolyte.
The electrons are urged through the external circuit by the
thermodynamic driving force called the Ciibbs free energy of the
overall reaction.
The major difficulty noted by Grove in 18W) is still a problem
in the design of fuel cells: how to obtain sufficient fuelelectrode
(catalyst)eleetrolyte reaction sites in a given volume. In many
Fig. 20.13 Unlike other conver
sion systems, fuel cells ore more
efficient at tow output.
Fuel cell
50 100
Rated bod, per cent
cells, fuel (gas), electrolyte (liquid), and electrode (solid) are
brought into effective contact by a porous electrode structure
which depends on surface tension forces to get reasonable contact
stability.
In theory a fuel cell can be built in almost any size and
capacity. Practically, fuel cells are packaged in small modules
or "batteries" to be connected in series or parallel as needed for it
particular application (Fig. 20.12). While conventional gener
ating devices hei ie less efficient as they so front design load to
idling, the fuel cell is more efficient at lighter loads (Fig. 20.13).
APPENDIX
Reaction Thrust
The concept of reaction Hi nisi may lie clarified by considering the
recoil produced by a parallel si ream of particles. Prom Newton's laws
it follows that for any system <>f objects or particles tin center of mass
of the syiem moves according to the equation
F = , mv
at
(1)
where F = net external force applied to system
m = total mass of system
v = velocity vector of the center of mass
/ = time
Xo matter how complicated the system or how inncli force one of its
parts exerts on any other, if the net external force is zero (as in field
free space), then
It™
il
(2)
which stales thai (he total moment urn of Hie system is a constant vector
iii:inl ity.
Consider a system of two particles, a "rocket" of mass m ami velocity
and a particle of (jas of mass 8m which is just leaving the nozzle with rcla
live velocity t.. The uel momentum of this system is mv + 8m{r — v,).
From Eq. (2)
j [mv + 6m (v — iv)] =
289
290 Appendix A
Rut 8m(di</dt) is negligible, and d(6m)/dt = ilm ill. since t he mass of
exhaust gas equals the decrease of the mikei max. Also for the exhaust
velocity r, . ilr, ill = 0, and m is a small quantity which approaches sera
in the limit. We have the result
ilr dm
m — = — v.
dl dt
(3)
or
lit
(4)
where F is the reaction force on the rocket. The mass flow leaving the
rocket dm/<lt is re presented by a positive Dumber. The negative sign in
the equation expresses the fact that Fand '•, are in opposite directions.
APPENDIX
B
Burnout Velocity and Range
In differentia] notation, Kq. (:i.bs) of A><. :i7 may be written
du rfflll m)
(i)
Kven if the thrust is not constant, this equation can be integrated to
give the velocity !>, at burnout
«i — i'o = ffu/, In h gtt, cos 8
flit
(2)
If we assume that the rocket starts from rest. c„ = 0, set /. = i.\. n "g,
and R = Wo/»i&, the ratio of initial mass t" final or burnout mass.
Kq, (2} may be put in the form
i'» = '"tr In R — gtt. eos 9
(3)
Here 6, is the duration of burning in seconds. The two averages f, (( and ;/
are necessary since the values of both effective exhaust velocity and
gravitational acceleration are dependent on altitude.
The altitude reached at burnout for a rocket in dragfree vertical
flight with practically constant thrust (dttt/dt = const) turns out to be
h = g„I,k f 1  p _ . 1 ~ il?«'i + <Vb + Ita
W
291
292 Appendix B
where Ao is the initial altitude at the start of burning. After burnout, the
rocket will coast upward to its maximum hoi^Iii h m , Again assuming
vertical nielli and negligible (has. hut taking into account the variation
of g with altitude during coasting, (lie masting distance is
A c =
>;■
(r. + ft*)'
2{?d r, 5  t»*(r, + h b )/2g tt
(«)
where r r is the radius of the earth. For a rocket which reaches a summit
of no more than a few hundred miles, h, is much smaller than r, ami
\±i\. (5) rednees to a familiar form
lu M
(6)
The summit altitude A„ reached in this vertical flight is
h m = A* + A c ^j
To approximate the range <>f a ballistic rocket, one may treat the
powered portion of the trajectory as vertical ami the coasting portion
a< elliptic. The coasting range a, along the surface of a nonrotating earth
has been determined as
 2r f sin 1
rf
2g„r,
**
(8)
The range calculation can be corrected for the earth's rotation by using
for i>i the vector sum of relative burnout velocity and the velocity of
the launching site and by adding veclorially to h l the distance the land
ing point move's while the rocket is in flight. If %» small, Bq. (8) reduces
to i he familiar equation for the range of an ideal parabolic trajectory on
a Hat earth:
0u
(9)
If r b is large, but less than (2ff r ( )i, t he denominator of Fq. (S) approaches
zero and s r becomes r f . Hence a burnout velocitv of (2ffur,.)J is just
sufficient for the rocket to enter into a circular satellite orbit.
The optimum angle of elevation * »f the trajectory at burnout
vanes with the desired range * r according to the relation
tan * 
1  sin (s r /2r r )
ooa (sJ2r,)
For short ranges, + = 46", For longer ranges, * is Jess than t.V
(10)
APPENDIX
Schrodinger Wave Equation
If * is the amplitude of the de ftroglic wave,
W W ** £ m
The X in this equation is to be found from the momentum of the particles
we are discussing. The momentum /' can be related to the kinetic energy
I mV
Ei _ ' ^ . JL lir p = V^B,
in
2m
<2)
The total energy B of a particle is its kinetic energy K, : plus its potential
energy E„, so
h a
~ p~ V2m"c¥^re7)
and the wave equation becomes
aso, xiy ^ity JU1
dx dy* ^^ 1 A*
(3)
(4)
E. Schrodinger showed, in I92(i, that Bohr's rules of quantisation could
be explained on the basis ( ,f the solutions of this equation. The quantity
ty (jisi) is called the "wave function" or the "probability amplitude."
Although + may be negative (or even complex), it turns out that its
293
294 Appendix C
We may ask what sort of eolations this equation would have for
foe electrons moving m the +x direction. Sinee no forces are applied to
theeteetrons, they move with constant velocity. Their potential energy
W the same at all pointe; we may take /' equal to zero. The solution of
he wave equation in tins ease will be a plane wave, expressive in terms
of Bmea and cosmos, just as for an electromagnetic wave.
APPENDIX
1. BOOKS FOR A PHYSICS TEACHER'S REFERENCE SHELF
American Association for the Advancement of Science: "The Traveling
Ilijdi School Science Library." AAAS and National Science Founda
tion, Washington, D.C., 1961.
American Institute of Physics: "Physics in Your High School," McGraw
Hill Book Company, Inc.. New York, I960.
Hrown, Thomas II. (edL): "The Taylor Manual of Advanced Under
graduate Experiments in Physics," Addison Wesley Publishing
Company, Inc., Heading, Mass., 1959.
Deason, 11. J. (ed.): "A Guide to Science Beading," The New American
Library of World Literature, Inc.. New York, 1963.
Glasstone, Samuel: "Sourcebook on Atomic Energy," D. Van Nostrand
Company, Inc. Princeton, N..L, 195K.
Hodgman. CI), (ed.): "Handbook of Chemistry and Physics," Chemi
cal Uubber Publishing Company, Inc., Cleveland, 1963.
[(niton, Gerald, and 1). 11. I >. Poller: ■■Foundations of Modern Physical
Science," Addison Wesley Publishing Company, Inc., Heading,
Mass.. ]!)oS.
Miehols. W, C. (ed.): 'The International Dictionary of Physics ami
Electronics," I). Van Nostrand Company, Inc., Princeton, N..L,
1956.
National Science Teachers Association: "New Developments in High
School Science Teaching." Washington, D.C.. I960. I neludes 9page
list, "Additional science program materials available."
Orear, Jay: "Fundamental Physics," John Wiley & Sons, Inc.,, Now
York^ 1961.
Parke, N. G.: "Guide to the Literature of Mathematics and Physics,"
Dover Publications, lue., New York, 1958.
295
296 Appendix D
Physical Science Study Committee: "Fhyaes," D. C. Heath and
Company, Boston, !%().
: "Laboratory Guide for Physics," J). C. Heath and Company
Boston, IWiO.
Price. Derek John deSollu: ■'.Science since Babylon," Yale University
Press, New Haven. Conn,, 1961.
Kesuick, R., and I). Halliday: "Physics for Students of Science and
Engineering," John Wiley & Sons. Inc., New York. I960.
Rogers, Eric M,: "Physics for the Inquiring Mind: The Methods,
Nature and Philosophy of Physical Science." Princeton University
Press. Princeton, N.J., HltiO.
Rouse. L J., and R. J. ISarllc: "Experiments for Modern Schools,"
John Murray (Publishers), Ltd.. London, 195(5.
Weber, R. I.., M. W. White. :iud K. V. Manning: "College Phvsies.'*
MeCrawHil] Hook Company. Inc. New York, 1959,
White. M. W., K. V. Manning, and R. L. Weber, "Practical Physics,"
McC rawHill Book Company, Inc., New York, 1955. Includes 33
experiments.
2. SOME PERIODICALS FOR A SCHOOL SCIENCE LIBRARY
American Journal of Physics. American Institute of Plivsies. 335 Last
45 St., New York 17. N.Y.
Xature, Macmillan & Co. Ltd., St. Martin's St., London, WC 2,
England, and St .Martin's Press, inc., 103 Park Ave., New York 17,
The Physic* Teacher. American Institute of Phvsies, 335 East 45 St.
New York 17, X.Y.
Physics Today, American Institute of Physics, 335 East 45 St., New
York 17, X.Y.
The School Science Renew, The Science Master's Association, 52 Bate
man St., Cambridge, England.
Science, American Association for the Advancement of Science, 1515
Massachusetts Ave., XW, Washington 5, IXC.
The Science Teacher, Journal of the National Science Teachers Associ
ation, 1201 16 St., XW, Washington (i, D.C.
Scientific American, (15 Madison Ave.. New York 17. N.Y.
Sky and Telescone, Sky Publishing Co.. Harvard College Observatory,
Cam I) ridge 3M, Mass.
3. SOME PROFESSIONAL ORGANIZATIONS OF INTEREST TO THE
PHYSICS TEACHER
American Association of Physics Teachers. American Institute.
Physics. 335 East 45 St., New York 17. N.I .
Am e rican Chemical Society. 1155 16 St., XW, Washington 25, D.C.
American Meteorological Society, 3 Joy St.. Boston S, Mass.
of
Appendix D 297
American Backet Society. 500 Fifth Ave,. Xew York 36, N.Y, (Ask for
latest Book List.)
American Society for Engineering Education, W. L. Collins, National
Secretary, University of Illinois, Frbana. 111.
Astronomical League, 310 Livingston Terr.. BE, Washington 20, D.C.
Commission on Mathematics, College Entrance Examination Board,
425 West 1 17 St., New York 27, N.Y.
Committee on School Mathematics. University of Illinois. Urbana, 111,
Educational Testing Service, Princeton, N.J. (The Cooperative Test
Division publishes a loose leaf binder, 805 pp., of Questions and
Problems in Science, Text Item Folio no. 1. 195ft.)
National Association of Biology Teachers, Paul Webster. Secretary
Treasurer, Bryan City Schools. Bryan, Ohio.
National Education Association, 1201 Hi St., NW. Washington (i, D.C.
National Science Teachers Association, 1201 16 St., NW, Washington ft,
D.C.
School Mathematics Study Group, Drawer 2502A, Yale Station, New
Haven. Conn.
Science Master's Association, John Murray (Publishers), Ltd., 50
Albemarle St., London, Wl, England.
Smithsonian Inst it uf ion, Washington 25, D.C.
4. SOME SUPPLIERS OF PHYSICS APPARATUS FOR TEACHING
Central Scientific Division, Cenco Instruments Corp., 1700 Irving Park
Road, Chicago 13. III., and (it Hi Telegraph Rd.. Los Angeles 22,
Calif.
The Ealing Corp., 33 University Rd., Cambridge 3S, Mass.
Macalaster Bicknell Co., 243 Broadway at Windsor St., Cambridge,
Mass. (Suppliers of PSSC apparatus.)
Science Materials Center. 5!* ['mirth Ave., Xew York 3, N.Y.
The W. M. Welch Scientific Co.. 1515 Sedwick St.. Chicago 10, III.
5. GREEK ALPHABET
A
a
alpha
N
V
nu
Ii
fi
beta
£
xi
r
7
gamma
omieron
a
a
delta
ii
TC
l'i
E
e
cpsilott
P
P
rho
Z
r
seta
V
a
sigma
H
n
eta
T
r
tau
n
1 liela
T
V
upsilon
1
I
iota
*
<P
phi
K
K
kappa
X
X
chi
A
X
lambda
*
*
psi
M
n
inn
Q
w
omega
298 Appendix D
6. SYMBOLS
= means equal to
■ means is defined as, or is identical to
^ means is not equal to
= means varies as, or is proportional to
2 means the sum of
* means average value of j:
« means is approximately equal to
> means is greater than "(» means much greater than)
< means less than (« means much less than)
" Voflln™ {hiir , ° V, ' P <!ifii,) '*&**** fiwt doubtful digit; e.g.
12,600 ik stated to only three signifieant figures: 1.20 X W
7. BRIEF INSTRUCTIONS ON USE OF A SLIDE RULE*
irtwL£?a&' ^P^H? 11 » «««*»% performed on the "C"
? I »i « S T caes ,? he " um! " ,r J " on the left end of the seale is ,, t , ( 
Su£%T££S! ^"lattherigluemlofthescalei!
flufe /or multiplication: Set index of "C" seale over cither „f th«
second fac or on the "C" scale. Read the answer on the •<!>" J..
under the ha,rh„e. Determine the location of the decimal point I, "i
rough mental approximation. ' ■
kxamplk: Multiply 17 X 23. See Kg. D.I.
(lit) Set Left "C" Index
Over V on "D" Scole
for
1 1 ■ ■> Tr
Fig. D.l
Life?.t,v3
—
a:
i:
:^r
^i t ..'.,i i ^i.: > ..;''  ' , i ii i ^ i >: t i Vt* ii i i Hbt
_.
(2nd) Under 23 on "C" Scole
Read 391 on "D" Scale
//ow (o Dtafe. Division j s generally performed on the "C" and "D"
scales also. "
the^^^utS'', 8 '' ^T 'T 1 ""' ° W tte » um ^tor (dividend) on
tin » scale and bring the denominator (divisor) on the "C" scale
gen*Co. StnU ' tbnS "'"' ' ,,)VriK!l1 iHu8 *«*ti«M <"">!esy of Eugene Diets
Appendix D 299
under the hairline. Head the answer on the "D" seale, under the index of
the "C" scale. Determine the decimal point by rough mental approxi
mation.
EXAMPLE: Divide by 3. See Fig. D.2.
<hr) Set 3 or." C"
Over 6 on "D"
r
2
c i : rt ' *? T V h'K 1
i i ■ u
1 1 1 iJi i
T
li I' *
lT
(2nd) Under Left "C" Index
Read 2 on "D"
Fig. D.2
flow to Find a 8quctre (too'.. Problems involving square roots are,
worked on the "A" and "B" scales in conjmietion with the "C" and
"I)'' scales. Note llial the "A" and "li" scales are divided into I wo
identical parts, which will he referred to as "Aleft" and "Afight."
Rule for square roots: If the number is greater than unity, and has an
odd number of figures before the decimal point, set the hairline over the
number on "Aleft" and read the square root under the hairline on the
"D" scale. If the number has an even number of figures before the
decimal point, use "Aright" instead of "Aleft." Locate the decimal
point in the answer by mental approximation.
If the number is less than unity, move the decimal point an even num
ber of places to the ri^ht until a number between I ami 100 is obtained.
Find the square root of the number thus obtained as explained above.
To locate the decimal point, move it to the left onehalf as many places
as it was originally moved to the right.
kxamclk: Find the square root of 507. See Fig. D.3.
(lsl) Set Hairline to 567
on Left Ho If of "A"
^
.c i.„.V4.r.Vi" ".?■%■'■&
o i 7i ■; ~i h I tTwWWl
TL
iiiiii jm ii i ii  iiiiii t} i»i nji i ; i liK
> } i : 1 1 1 it i t f 1 1 1 : [ w ; J i w jhii  hiih 4 hii 1 i ^ ■ »
i),.  i l "
\s~rs
(2nd) Under Hairline on
"D" Read 238
Fig. D.3
300 Appendix D
l*sc "A4aft," since than is mi odd number of figures before the
decimal point. By mental approximation, locate the decimal poinl
after the second significant figure, making the answer 23.8.
kxamplk: Find the square root of 0.0956. See Fig, ]>.].
(1st) Set Hairline Over
9.56 on Left Half of "A"
* '■■•■'•■.' ...r W..I., .t,„. > . t . >
D I .1 ■ .1 ■ .1 n n'~a •> n'TT) ~
~m
• • < ■ "■!" ),...i...*..i...j . « p j , t.i.ip
,k • O im]..l,,.Jl^,l,ll. ( .. , 1( , ,ii m i; ■
(2nd) Under Hairline on
"D" Read 3.09
Fig. D.4
.Move the decimal point two places to the right, thus obtaining 9.56.
Fse the "Aleft," hecause there is now an odd number of figures before
the decimal point. Take the square root of 9,r>6. !hen move the decimal
poiul one place to the left, making the answer 0.309,
7A. SLIDE RULE BIBLIOGRAPHY
Bishop, C. ('.: "Slide Rule How to Fse It." Barnes & Xohle, Inc..
New York.
Bshbaeh, 0. \V., and H. L Thompson: "Vector Type Log Log Slide
Rule." .Manual no. 1725, llugene Dietisgen Co., 1009 Vine S(
Philadelphia 7. Pa.
Naming. M, L:"A Teaching Guide for Slide Rule Instruction," Pieketl
and Eckel Inc., 1109 South Fremonl Ave,, AJhamhis, Calif
Harold, Don: "Slide Rule? May I Help . . . ," KeulTel and Baser Co.,
Adams and Third Sis., Hohoken, X.J.
"Inirnducing the Slide Rule," Wabash Instrument & Specialties
Company, [tic. Wabash, lud., 1943.
"Ii' Easy to Use Four Post Slide Rule." Educational Director, Freder
ick Post Co., 3050 North Avondale Ave., Chicago, 111. (A projec
tion slide rule is expected to be available soon for classroom use.)
Johnson. L. H.: "The Slide Rule," I). Van Xostrand Company. Inc.,
Princeton. X..1.
Macliovina. P. K.: "A Manual for the Slide Rule," McGrawHill Book
Company, foe, 330 Weal 12 St.. New Ymk 30, N.Y., 1950.
"Mathematics, .Mechanics, and Physio." Engineers Council for Pro
fessional Development, 29 West 39 St., New York, N.Y.
8. TRIGONOMETRIC FUNCTIONS
Radians
[;•■:.]'.•.■'.
Sine*
Cosine?
Tang ants
Cotangents
.0000
a
.0000
1.0000
,0000
s
00
1 . 5708
.0178
i
.0175
,9988
.0175
57.20
80
1 . 5533
.(Will
2
.0349
.9984
.0340
28 . 04
88
1 ,5359
0524
if
.0523
.9986
.0521
10,08
87
1.5184
,0698
4
.0008
.9976
.0699
14.30
86
1 5010
.0878
5
.0872
.9962
.0875
11.430
85
1 . 1835
Ml 17
. 1046
.9946
.1051
n 61 i
84
1 . (661
. 1222
7
1219
. 05)25
. 1228
8.144
83
1.4480
. 1390
8
.1392
mod3
.1 105
7.115
S2
1 .4312
.l, r >7]
9
. 1504
.9877
. 1581
0.314
81
1.4137
.1745
10
.1736
.9848
. 1703
5.671
SO
I 3963
.1920
11
ions
.9810
.1914
5 ! 15
79
1.3788
.2091
12
,21)70
.978!
.2120
4.705
78
[ .3614
2209
13
.2250
.9741
.2300
4.332
77
1.3439
21 !3
14
.2410
.9703
.2493
I Oil
70
1.3205
.2(il8
15
2588
0059
.2070
3.732
75
1.3090
.2798
16
•J 7511
.9613
.2867
3.487
74
I .2918
. 2967
17
.2021
, 9563
.3057
3.271
73
1.2741
.3142
18
.3090
.9511
.3240
3.078
72
1 .2666
,3310
19
.3250
.0455
,3443
2 904
71
1.2302
.3491
20
.3420
.9397
.3640
2.748
70
1,2217
, 3605
21
3584
.9336
.3839
2.606
69
1.2043
. 38 H>
22
.3746
9272
.4040
2 175
08
1.1808
. 101 1
23
.3907
'.1205
. 4245
2.350
67
1.1694
.1189
21
.4007
.9136
.4152
2.246
«a
1.1519
. 4363
25
.4220
.9003
.4003
2 144
05
1.1345
.1538
211
.4384
.8088
,4877
2.050
< 1
1.1170
.1712
27
. 4540
.8010
. 5095
1 . 963
68
1 .0990
.4887
2S
1695
.8829
.5317
1.881
02
1.0821
.5081
29
IMS
.8740
.5543
1.804
01
1. O047
5236
30
.5000
. 8660
.5771
1.732
00
1 .1)172
.5411
31
5150
S572
.0009
1.664
59
1.0297
.5586
32
.5299
S1S0
0249
1 .000
58
1.0123
.5700
33
.5440
.8387
.0494
1 54(1
57
0.9948
. 5934
34
5592
.8200
. 0745
1 is:;
56
0.9774
Jill)!)
35
.5736
.8102
7002
1.428
55
0.9599
.6283
: j ,r,
.5878
SOOt)
7266
1 370
54
(1 1)125
.0458
37
.0018
.7980
. 7530
1.327
53
0.9250
.6632
38
11157
7SS0
.7813
1 . 280
52
0.9076
.0807
39
.112!):!
.7771
.8098
1.235
51
o 8901
.0081
40
lltiS
. 7660
. 839 1
1 . 192
50
0.8727
.715(1
II
.0501
.7547
8693
1 . 150
49
0.8552
. 7330
42
.0091
.7431
.9004
I. Ill
48
0.8378
. 7605
43
.0820
. 73 1 1
,9325
1.072
47
0,8203
. 7079
44
.0047
. 7 1 93
.0057
1.030
46
0.8029
. 78, r )4
45
.7071
.7071
1 0000
1.000
45
0.7854
Coiinei
Sinfti
Colon genii
Tongont?
Degree*
Rudiufis.
301
302 Appendix D
9. LOGARITHMS TO THE BASE s
These two pages give the rial lira! (hy
iierbohe, or Napierian) logarithms of tnnii
bera between I and m, correcl in four
places. Moving the decimal point » places
to the right (or left) fa tfie number is
equivalent to adding ,, limes 2,:<02(i (or n
tunes 3.6974) to the logarithm.
1
'2
3
•1
5
6
i
8
2.8026
i 6052
6 B078
8.2103
11.6129
13.8165
IS 1181
18.4207
20.7233
I
2
3
4
G
6
7
8
9
1.0
i.i
i.j
i. a
1.4
l.S
1.6
1.7
l.S
1.9
2.0
2.1
J.J
2.3
J.4
J.S
2.0
J.7
2.8
2.6
3.0
3.1
3.2
J.S
3.4
3.5
3.0
J.7
3.8
S.9
M
4.1
4.2
4.3
4.4
4.5
4.0
4.7
4,6
4 LI
0.00OQ
0953
1823
1624
3305
4055
470(1
5300
5S7.V
0419
0.8931
7419
7885
8329
8755
9ISJ
9555
0.9933
1.0296
0647
0100 0108 iiair, 0392
1044 1133 1222 1310
IMS J070 2151
2770 2852 2927
mm
2700
1314
1632
10811
•>•>$<■
2528
1809
3083
3350
JO 10
1.3883
4110
436!
!>,,
4810
5041
6201
5476
5688
6892
1436 3607 3577 3640
4121 4187 4253 4318
4762 4824 4836 4947
53S5 5423 6481 SS39
5933 5988 8043 6098
8471 0523 6675 6627
698t 7031 7080 71JS
7467 7514 7561 7608
7930 797S 8020 8065
S372 8416 8450 850J
8700 8838 8S70 8020
9203 0243 92*2 9322
9504 9632 0670 9708
9960/0006 0043 0080
033J 0367 0403 0438
0682 0710 0750 0784
1019 1053 1086 1119
1346 1378 1410 I44J
1063 1094 1725 1756
1009 2000 2030 2060
2267 2206 2320 2355
2556 2585 2613 2641
2137 2S«5 2M)2 2M
3110 3137 3164 3191
3376 3403 3429 3455
3635 3661 3680 3712
3888 3013 3938 3962
4134 4159 4183 4J07
4375 4398 4422 4446
4600 4633 1666 4670
4839 4861 4884 4907
5063 5085 5107 5120
5282 5304 5320 6347
5497 5518 5530 5560
5707 5728 5748 5769
6013 5033 5953 6974
0488 0583 0077 0770 0862
1398 1484 1570 1055 1740
2231 2311 2390 2469 JS46
3001 3075 3148 3221 3293
3710 3784 3853 3920 3088
43S3
5008
.i,W,
6152
6678
7178
7665
8109
8644
8901
9361
0746
0111,
0473
IIS IS
1161
1474
1787
2000
J3S4
2669
2947
3216
3481
3737
3987
4281
4469
4702
4929
51SI
6309
5581
5790
5994
4447 4S11 4574 4837
5068 5IJ8 SI8S 5J47
56.53 5710 5766 5822
6208 0259 0313 0366
0729 0780 6831 6SS1
7227 7275 7324 7372
7701 7747 7793 7839
8154 8198 8242 8286
8587 8829 8671 8713
0002 9042 9083 9123
9400 9439 9478 gsi7
9783 9821 98S8 9806
0152 0188 0225 0260
0508 0543 0578 0513
0952 0886 0918 0953
1184 1217 1249 1282
1506 1537 L500 1600
1817 1B48 1878 1009
21 19 2149 2170 JJ08
24 13 J442 2470 2499
MM
2976
3244
3507
3702
4012
4255
4403
47J5
4!>. r ,l
5173
6390
5602
S8I0
6014
2720 2754 2792
2002 3020 3056
3271 3297 3324
3633 3558 3584
3788 3813 3838
4036 4001 4085
4279 4303 4327
4510 4540 4563
474s 4770 471)3
4074 4090 5019
5195 5217 5230
5412 5433 5464
5023 5844 5865
5831 5851 5872
8034 8054 8074
19
0.0053
1823
2024
3365
4055
4700
5306
5878
6419
0.6931
7419
7885
8320
8755
9103
9565
0.9g33
1.0296
0647
1.09S6
1314
1632
1939
2238
2526
2800
3083
3350
3610
1 3m;;i
4110
435)
4586
4816
6041
5261
5476
5680
6802
1.6094
69743
'.HUH ."i
05)227
789710
487112
I Sir, I I
881917
679319
27(1721
Tflnrhf of the
Tabular
Difference
12 3 4 5
10 10 29 38 48
17 20 35 44
8 15 24 32 40
7 15 22 30 37
7 14 Jl 28 34
13 19 26 32
8 12 18 24 30
8 II 17 23 J8
5 II 18 22 27
5 10 15 21 26
5 10 15 20 24
5 9 14 19 23
4 9 13 IS 22
4 9 13 17 21
4 8 12 16 20
8 II 10 20
8 II 15 19
7 11 15 18
7 It 14 18
7 10 14 17
7 10 13 18
6 10 13 16
6 II 15
6 9 12 15
6 9 12 14
8 II 14
8 II 14
8 II 13
8 10 13
8 10 13
2 5 7 10 12
2 5 7 10 12
2 5 7 II
2 5 7 9 11
2 4 7 9 11
J 4 7 9 II
2 4 6 9 II
J 4 8 8 11
J 4 8 10
2 4 6 8 10
Appendix D 303
LOG, (BASE e = 2.718284)
Tenrht
of the
Tabular
Difference
9
1 I
1 4
5
6 7 8
9
10
1234 5
5.0
1.8094
0)14 6134
0154 0174
6194
0214 6233 6263
0273
6292
2 4 6 8 10
G.1
112! 12
8311 6332
6351 6371
6390
8409 8429 8448
6487
6487
2 4 6 8 10
8.2
6487
6508 66J5
6544 6563
8582
6601 6620 6630
6668
6677
2 4 6 8 10
S.J
6677
0696 0715
6734 0752
6771
6700 6808 6327
6845
6864
2 4 6 7
6.4
6864
8882 8901
6910 6938
MM
6974 8993 7011
7029
7047
14 6 7 9
6.5
7047
7060 7084
7102 7120
7138
7156 7174 7192
7210
7128
24 57
6.6
7228
7246 7J63
7281 7199
7317
7334 7351 7370
7387
7405
2 4 5 7 9
6.7
7406
7422 7440
"457 7475
7492
7609 7527 7544
7561
7579
2 3 5 7 9
6.8
7679
7690 7813
7630 7647
7864
7681 7099 7716
7733
7760
2 3 5 7 9
5.9
7760
7760 7783
7800 7817
7834
7851 7867 7884
7901
1.7618
2 3 6 7 8
CD
1.7016
7934 7951
7967 7984
8001
8017 8034 8050
8060
8083
2 3 6 7 8
6.1
8083
8090 8116
6132 8148
8185
8181 6197 82)3
,j,,l
8245
2 3 5 7 8
6.2
6245
8262 8178
8294 8310
8326
8342 8358 8374
8390
41'),
2 3 5 8
6.3
8405
8421 8437
8453 8460
S4S",
8500 8516 6532
8647
8563
2 3 5 8 S
6.4
8563
8579 8594
8810 6025
8041
silSft K67J v ( ;s:
8703
8718
13 5 6 8
6.5
8718
8733 8749
8784 8779
8795
8810 8825 8840
MM
8871
2 3 5 S
6.6
8S71
8886 8901
8918 8931
8940
8961 8976 8991
9008
0011
2 3 5 6 8
0.7
9021
9036 0051
'.Win; iwii
0095
9110 9125 9140
9155
9109
13 4 7
6.8
0109
9184 9100
9213 9228
9142
9257 9272 9280
9301
9315
13 4 6 7
6.9
0315
0330 9344
9369 9373
93S7
9402 9410 9430
9445
1.9459
13 4 7
T.O
1.9459
0473 948S
9502 0516
0530
9544 9560 9573
9587
9601
13 4 6 7
7.1
96UI
0615 0629
0643 0857
0671
0686 9600 9713
9727
0741
13 4 7
7.1
9741
9755 0769
emg »th
0810
0824 0838 9851
9385
1 ',7'.'
13 4 8 7
7.3
1J870
9802 9900
9920 9933
9947
998) 9974 0B88/0O0I
2.0015
13 4 5 7
7.4
2.0015
0028 004!
0055 0069
0082
0000 0109 0122
OI30
0149
13 4 5 7
7.5
0149
0102 0176
(1I.1SI B202
0215
0229 0241 0255
0268
0281
13 4 5 7
T.fl
0281
0295 0308
0311 0334
0347
0360 0373 0386
0399
0412
13 4 5 7
7.7
0412
0425 043K
0461 0464
0477
0490 0503 0516
0528
0541
13 4 5 1
7.8
0541
0554 0567
0580 0592
0805
0618 0831 0643
0656
0660
13 4 5 6
7.9
IlW'.i
0681 0604
0707 071!l
0732
0744 0757 O70II
0762
2.07114
13 4 6 6
1.0
2.0704
0807 0810
0832 0844
0867
OS00 0882 0894
0906
ll'.llll
12 4 5 6
8.1
0010
0031 0843
0058 096s
0080
0002 1005 1017
1020
1041
12 4 5 6
S.2
1041
10S4 1080
1078 1090
1101
1)14 1126 1138
1150
1163
12 4 8 6
8.3
1163
1175 1187
1199 1211
1223
1235 1247 1158
1270
1282
12 4 5 5
8.4
1282
1204 1300
1318 1330
1342
1353 1305 1377
13SH
1401
12 4 6 6
8.S
1401
1412 1424
1436 1448
1469
1471 1483 1494
1506
1618
12 4 5 6
8.6
1518
1529 1541
1562 1564
1576
1587 1590 1610
1822
1623
12 3 5 6
8.7
1833
1645 1660
1668 16711
1601
1702 1713 1726
1730
1748
12 3 5 6
J .8
1748
1759 1770
1781 1793
1804
1815 1827 I83S
1849
1861
12 3 5 6
8.9
1861
1872 1883
1894 1906
1917
1028 1039 1050
1961
2.1972
12 3 4 8
J.O
2.1971
1083 1904
2006 1017
2028
2039 2050 2061
2072
2083
12 3 4 6
9.1
2083
2094 2105
2116 2127
1138
2148 2159 2170
2181
2102
12 3 4 5
1.2
1192
2103 2214
2225 2235
2240
2257 2288 2270
2 2 Mi
2300
12 3 4 5
9.3
2300
2311 2322
2332 2343
2354
2364 2376 2386
2396
1407
12 3 4 5
9.4
2407
2418 2428
2439 2450
2460
2471 1481 2492
2502
2513
12 3 4 5
9.5
2513
2523 2534
2544 1.555
2565
2576 2586 2597
2607
2616
1 2 3 4 F
9.8
2'Us
2628 2638
3649 21159
2670
2680 209O 2701
2711
2721
12 3 4 6
S.7
2721
2732 2742
2752 2702
2773
2783 2703 2803
2814
2v>4
12 3 4 5
0.8
2824
IS34 2844
2854 2865
2875
2886 2895 2906
1915
2 '.■:•.'.
1 2 3 4 S
9.9
2925
2935 1948
2966 1966
2978
2980 2996 3006
3016
2.3026
12 3 4 5
304 Appendix D
10. VALUES OF PHYSICAL CONSTANTS
As experimental data improve, "best values" of the physical constants
are recomputed by statistical methods. See, for example, K, It, Cohen,
J. \V. M. Do Mond. 'I". \V. Lay Ion, and J. S. Hollelt. "Analysis of
Variance of the 1052. Data on the Atomic Constants and a New Adjust
ment, 1885" Review of Modern i'hysvs, 27:303 380 (1955). The values
listed below have been rounded off from those liste<l in the paper cited
and have been expressed in inks units. The physical scale is used for
all constants involving atomic 0)88868.
Avogadro's number: A'.i = 6.0249 X 10 10 molecules kmole
Gas constant per mote: R« = 8,31 7 joules/(kmoie)(°K)
Standard volume of a perfect gas: V a = 22.420 m 3 atm. kmole
Standard atmosphere: i>» = 1.013 x 10* newtons/m*
Speed of light in free space; c = 2.9979 X 10" m/sec
Electronic charge: e = 1.0021 X 10 !l coul
Planck's constant: A = 6.6252 X SO" 3 " joulesec
Faraday constant: F = 9.652 X 10 7 coul/ kmole
Charge/mass ratio for electron: e/m = 1.758" X 10" coul/kg
Rest mass of electron: m = 9.1083 X 10~ 31 kg
First. Uohr radius: «„ = 5.2917 X 10 ll m
Compton wavelength of electron: X = h/mc = 24.203 X 10 13 m
Boltamann's constant: A. = 1.3804 x 10 S3 joule, °K
= 8.617 X10 B cv/"K
.'•lassenergy conversions: I kg = 5.610 X 10 s * Mev
1 electron mass 0,51098 Mev
1 proton mass = 938.21 .Mev
1 amu = 931.14 Mev
1 neutron mass = 939.51 Mev
Energy conversion factor: I ev = 1.6021 x 10 "joule
Rest masses: electron m = 9.1083 X 10 31 kg = ft.4870 X 10 • amu
proton »t„ = 1 .0724 X 10"" kg
neutron »i„ = 1.6747 X 10  " kg
Proton mass electron mass ratio = 1,830.12
(iravitational constant: = 6.67 X 10 ll newtonmVkg*
Appendix D 305
11. CONVERSIONS OF ELECTRICAL UNITS
Quantity
Symbol
Practical
unit, mfcs
Cgsesu equiv.
Cgsemu equiv.
Energy
Current
Electronic potential
Electronic field
Magnetic flux
Magnetic induct.
Permittivity of free
w
(
V
£
a
B
1 jouJe
1 ampere
1 volt
1 volt/m
1 weber
1 weber /m 2
BBS X 10" 11
coulV
newton ni !
1.257 X 10" 6
newton/
omp
10" ergs
3 X 10 9 jtotomp
J X 10 s statvolt
10~ 4 iv/cm
statcoulomb
dyne cm 5
1 1
9 10" V*wso
10" ergs
0.1 abomp
10* ob volts
1 f ' abv/cm
10^ maxwells
10' gauss
1 1
space
Permeability of free
space
9 ■ 10" V«M>
unll pole
dyne cm 5
Note: / —
= C
Index
Absorption coefficient, 265
Acceleration cine to gravity, 41, 43
Acoustic waves, 1(11, 103
(See a/so Wave)
Activity, 240
specific. 242
Adams, ('. C, 7. 8
Actlu i theory, 139, 1st;
Ampere, 110
Ampere's law, 131, 150, 161
Amplitude, !l(i, 106
Angle of ascending node, 51
Angle of inclination, 51
Angular momentum, 55
Anode, 140
Apogee, BJ
Apparatus, suppliers of, 297
Argument of perigee, 53
Wending node, angle of, 51
Asteroids, 10, 13
Astronautics, bibliography of, 7
careers in, 7
lilms on, 7
history of, 3
Astronomical unit, 9
Atmosphere, entry of, 72
Atom, 7!i
earlv concept of. 78
models or, 117. 203 205. 23\ 25:1
radius of, K5
speed of, 86
Atomic Knergy Commission, 32
Atomic muss, 70, 153
Atomic mass unit, 70
Atomic number, 118, 183, 205, 211,
220
Atomic weight, 79
Auger effect, 175
307
Autoradiography, 203
Average life, 24 1
Avogaclro's number, 78, 144
Baker, K. 11.. IS, 19
Bnlmcr Series, 208, 212, 214
Hand theory of conduct inn. _M">
Harkla, C. G, 281
Barrier, energy, 220, 244
Bertie, R. J., 290
Bauer. 0. A.., IS
Beats, 197
BecquereJL H. A., 77, 203
Benson, O. ()., 7
Bernoulli's theorem, I'll
Beta decay, 210
Bel si spectrum, continuous. 24S
Binding energy, 299, 242. 251
Bishop, C. ('.. 390
Blaekbody, 168
Blackbody radiation, 166
energy distribution in. lit)
(See also Radiation)
Blunchard. C. II.. 153
Bohr. N.. 1 18, 293
Bohr atom model, 293 204,297.222
(See iitao Atom)
Hook list. 205
Bore, 08, 9!)
Hot he. W.. 253
Bragg diffraction, electron analogy
of, 21 S
Breeder reactor, 271
Breillat rahluug, 17li
Brown, T. IS., 295
Bubble chamber, 203
Budierer, A. II.. 198
Bnchheiin, It. YV., 7, 70
308 Index
Burnett, C. It., 163
Burnout .speed, rocket, 20, 201
{See alxo Rocket )
Cajori, !■'., 64
Calorie, ss
Careers in astronautics, 7
[See also Astronautics)
Curncit efficiency. 32. 2x4
Cathode, 140 *
Cathode ray tube, 143
Cal lioiie rays, 1 30
Cavity radiation, L89
Center of mass, 20
^ rotation about, 212
Chad wick. J., 253
Chain reaction, 274
Charge, of an electron, 145
Charge /mass ratio, 127. 142
Charging, 117
Chromosphere, II
Circular orbit, 46, 53
Clock paradox, 195
Cloud chamber, 263
Cohen, ]■:. H., :«)4
Collisions, molecular, 86
nuclear, 252
Comets, 11), i:j
Common ell erf, 176, 265
Condon, E. U., 210
Conduction of elect rieity in a tas
139, 140
Conductivity, band theory of, 235
electrical and thermal. 233
of metals, 146
quantum theory of, 234
Conic orbits, 65
Conic sections, 64
Conservation of energy, 200, 244
Constants, physical, :mm
Corona, 1 1
Correspuiidence principle, 183, 214
J osmic rays, II, 14, 208
Coulomb. I It)
Coulomb barrier. 220. 244
Coulomb's law, lis, 200
Cou titer, 201
(leigerMuUer, 261
scintillation, 202
Crew, II., mi
< 'fit teal mass, 27 1
Cross product, vector, 56
Curie, 242
Current, conventional, 127
direction of, 12s
electric, (27
electron, 127
induced, 136
in magnet ie field, 130
in metals, Hti, 235
Daltnn, J.. 77. 7.s
Damped wave, 106
Darwin, C, R. 115
Davidson, U'. ].., 273
Davisson ami Germer cxiariimnt
2I.S
Deason, H. J., 205
De BrogUe, I,., 216
De Broglie wave, 170, 222, 230
Decai constant, 211, 25<J
Deflection of charged particles, 125,
Degree of freedom, S8
I 'elector, radiation, 2til
Do ^'auvenargues, Marquis, 138
Dewey, John, 40
Dill ruction, 107, ISO
of electrons, 2 IS
of neutrons. 210
of photons. 2 IS
Diffraction grating, 108
Dilat atio n of lime, I it
Dirac, P. ,\. M., 216
Direction rules, for induced emf,
136
for magnetic force, 134
for magnetic induction, 134
Disintegration energy, 243
I Kspucement, 06
Disraeli, Benjamin, lis
Distances, to planets. HI
to stars, IS
Dobie, .1. [''rank. ti7
Dopplcr effect, 112, 107
transverse, 1 14
Dose, radiation, 267
Dunne and Hunt law, 170
DnClaux, lit)
Dulongand IViit law, I4S
DuMoml.,1. U\ M., :«)4
Duncan, .1. c. mi
Dyne. 21
Earth satellite, fit
Eccentricity, .",1
Eddington, A. 8., 115. 271
Edison, T. A„ 2X1
Effective exhaust velocity. 25, 30
Efficiency of heal engine', 32. 284
Einstein, A., !t, 67, 77, 173, 175,
1X4, 104, 196
Einstein's massenergy equation,
199
Einstein's photoelectric equation,
173
ESectiio current, 127
in magnetic Meld, 130
in metals, 148, 235
Electric field intensity, [21
Electric potential, 122
Electrification, 1 17
Electrolysis, I It
Electromagnetic wave, 103, 161
energy of, (66
gamma ray, 242
plane, 103. (66
speed of. 104
xray, 175
Electron, 10s. 127, 130, 144
charge or, 145
and electrolysis, 111
e/m ratio for, 142
energy data for. Hi!)
free, 146, 232
in nucleus. 246
shells, 227, 230
Electron configuration, 228, 230
llecfroti (iilTraetion, 2 Is
Electron (low, 127
Electron theory of conduction, 146
Section volt, 124
Hectrostatic units, 1 19
demerits, periodic table of, 228
Elements of an orhil, 51
Jliot. C, 166
jllipse. 61, 64
Emerson, It. W., 77, 1S3
imf, induced, 135
Energy, binding. 20(1, 254
conservation of. 21 Ml, 244
disintegration, 243
distribution in blacklmdy radia
tion, 170
in electromagnetic wave, 105
oquipnrtilion of, SS
forbidden, 237
in nuclear reactions, 243, 25S'.
273. 270
potential, 44
quantization of, 2(H), 220
of satellite. 1 1
sources of, 272
tiergv barrier. 220, 244
uerg'y levels, 2IIX. 21(1, 235. 242
qua! ion of state. 81
quilihrium, radioactive, 250
quipnrthinn of energy, 8S
scape speed, 31, 45
shlmch, O. W., 301)
xelusion principle, 227
Ixlinust velocity. efTeclive, 25, 30
Exponential law of decay, 230
Falling body, 42
Faraday, ML 136, 144, 150, 283
Faraday, 144
Paradays law of induct ion. 158,
Kit I
Fermi, K., 273
Fermi distribution, 234
Index 309
field, electric, 121
deflection of particles bv, 125,
Ml
gravitational, 42
magnetic, 125
deflection of particles by, 125,
120, 141
Field intensity, electric, 121
Field strength, magnetic, 105
Film lists, 7, IS. 37
Fission, nuclear, 273
FitzgeraldLorents contraction, 104
Flux, electric, 157
magnetic, 135
Forbidden energies, 237
Force, gravitational, 41
magnetic. 134, 142
on a current, 120
nuclear, 206, 254
Fourier series, 07, 104
Founder. (I,, 201)
Franklin, YV. S., 138
Free electrons, 140. 232
Free fall. 43. 4li
Freedom, degree of, 88
Frequency. 114
Fuel cell, 284
Fusion, unclear. 274
a, gravitational acceleration, 41, 43
(7, gravitational constant, 41
Galaxies, 17
Camilla decay, 242
Gas, fully ionized. 275
ideal, 80
kinet ic I henry of, S!
Gas constant, so
Gas discharge tube, 140, 141
i las law, so
Gauss, 120
Gauss's law, 157. 150
( lavIaissac, .1.. 7s
Geiger. H., 203
(ieigerMnller counler, 261
rator principle, 186
Glaser. D. A,, 263
(ilasstone. S., 205
Goddard, H. II., 4
Gram tuoieciilar volume, 70
Grating, optical, 108
Gravitation, universal, 41
Gravitational acceleration, i/, 41
on planets. HI
slandard, 25
Gravitational constant, 0, 41
( iravitat iunal field. 42
Gravitational force, 41
on planets, 10
Gravitational potential energy, 44,
47
310
Index
Gravity. 41
( Irsek alphabet, 207
Croup velocity, l(J(l
Croups, orbital, 227, 230
drove, W., 284
(iuidnnco of rocket, "(I
Halm, a. 273
Halflife, 241
(See (i/.to Radioactivity)
Hall effect. 233
Ilalliday, I).. 2!lli
Hartung. M, I„. 300
Heat engine, ellieienev of, 32
Ilcavisido, (I., 120
Heiscubcrg, W,. 2 It;
Hciscnljerg a uncertainty principle,
182
Henry, J.. 135
Herald, ]>., 300
H err iik. K., 02
Hertz, (i.. 157. KM
Highejicrgv particles. 260
Hohbs. M„ 3S
Hodgroan, (". 1),, 2!>f>
Hull on, (i., 205
Horace. Ill
I lo vie, V., I!)
Hubble, !■:. P.. go
Huxley. T. H., 252
Hydraulic jump, SIM
Hydrogen, Hulir model of, 207
energy levels for, 208, 210
spectrum of, 207
(See alao Atom)
Ideal ga« law, SO
I (u puke, specific, 25, 27, 32, 33
Inclination, angle of, 51
Induced current, 135
direction of. 135
Induced end. 135
direct ton of, 138
Induction, magnetic, 128, 131
at center of loop. 132
direction rule for, 134
Faraday's law of, IjjK
force due to, 134
near Straight wire, 133
Insulator, 237
Intensity. ]{I5
wave, 165
(See alto Wave)
Interference. 106, 1*7
Interferometer. iMi
Interplanetary travel, OR
Ion. 144. Mi)
Ion propulsion, 34, 3d, 71
Ionization chamber, 201
Ionization energy, 229
Ionization potential, 2in, 211
Ionizing radial ion, 2(51
iHHtojws, 140, I />:*
Ives. II. i:.. Ml
Ives and Stilwell experiment, li)7
James, J. N., m
Jet .separation, 2!l
Johnson, L. 11., 300
Joliot. 1''.. 253
Jupiter, lit, 12
A*eapture, 217
Kelvin, Sir William Thomson, 271
Kepler's laws, 56, (HI
derivation of, (i2
Kinetic theory of gases, 81
Kirchhoffs law, lli.s
Kiwi engine, 32, 33
Krogdahl, W. S.. I!)
li ert, B., St!. S7
Launching speeds, 70
(See also space)
Layton. 1'. W., 304
Lens's law, 135
Life, on planets, is, 10
of radioisotope, 241
Light, speed of, (>4, 8(i, 1JI2
Line of force, gravilat imial, 42
Loeb, L. I).. s:i
Logarithms, 3112
Lorents, H. A., 232
Lorentz transformation. 193
Much number, 2s, 102
Machoviaa. !'. K.. 300
McLaughlin. !l. ».. Ml
Magnetic deflection, I3ti, 141
Magnetic field strength. I(i5
Magnetic Mux, 135
Magnetic induction, I2K, 131
ill center of loop. 132
direction rule for. 131
Faraday's law of, 158
force due to, 134
near straight wire. 133
M.'ignclohydi'odyiiariiics, 34, 283
Manning, K. V.. .".Id
Marconi, (!., 157
Mars, HI, 12
Marsdon. E., 203
Mas.s. atomic. 70
niolcnilar, 7'.)
Massenergy relation, HIS
Mass number, I IS
Mass rat io, 30
Mass spectrometer, 131, Mil
Uainbridge type, 152
Dempster type, 151
Index 311
Matter, composition of. 117
four states' of. 27(i
Matter wave, 170, 217, 230
Maxwell..). C. 83, 87, 88. 103, 166,
lid. 1114
Miixwelliloltzinann distribution,
s:{. 87, 232
Maxwell's electromagnetic theory,
186, Mil. 107
Maxwell's equations. Mil
Mean live path, 84
Mechanics, principles of, 21
Mendeleev, 1). 1., 228
Mercury, HI. II
Mesons, 107
Metals, conduction in, 232
Meteorites. HI, 13
and radio waves, 13
Miehels, W. ('.. 205
Micbelson interferometer, lst>
MichelsonMorlev experiment, ISS
Slilky Way, 17
Miliicurie. 242
Millikan, H. A.. 145
Mills, M. M, 38
Missilery, chronology of, 4
Model rockets, 37
manufacturers of, 37
(Seeaiso Rocket)
Models of molecules, 8(1
Modern physics, 77
Mole, 79
Molecular mass, 7'.i_
Molecular volume. 70
Molecule, 70. 117
mode! of. 88
Momentum, angular, 55
quantisation of, 209, 220
in relativity theory, 198
Moon. HI, 12
Moslev's (aw, 231
Motion. Newton's laws of, 21, 43
uniformly accelerated, 22, 42
Motors, rocket. 27. 32
(See alxo Hockot 
NASA (National Aeronauties and
Space Administration), 7, 8, 32
National Association of Rocketry,
37
Mangle, J. E., 14, 17. 19
Neptune, 10, 12
Neutrino, 240
Neutron, 118, 274
detection of, 263
discovery of, 253
Newell. H. K.. 14, 17. HI
Newton, Isaac. (HI
Newton (unit), 21
Newton's law of gravitation. II
Newton's laws of motion, 21, 43
Node, 51
Nonionizing radiation, 261
Nozzle, rocket, 2S, 33
(See also Hocket)
Nuclear atom model, 205, 238, 253
Nuclear landing energy, 200
Nuclear emulsions, 203
Nuclear fission, 273
Nuclear force, 206, 254
Nuclear propulsion, 31, 71
Nuclear reactions, energy from, 243.
25S. 273. 270
equations for. 25ti
threshold energy for, 258
Nuclear reactor, 32, 273, 274
breeder, 274
Nuclear testing, 269
Nuclear wastes, 270
Nuclei, stable, 255
Nut ■icon. 1 17, 254
Nucleus, 1 17, 230
radius of, 254
Oersted, B.C., 110
Orbit, circular, Hi, 53
electron, 227,230
for entry o[ atmosphere, 72
in hydrogen atom. 2111), 222
satellite, 47
conic, 66
elements of, 61
precession of, 57
Ordwnv. I 1 '. L. *
Orear, .1., 63, 296
Oslwald, W., 281
Page, I.., S3
Pair production. 243, 266
Pairing of aueleons, 255
Parke, X. <■',., 296
Pauli exclusion principle, 221
Performance of liquid propellents,
20
Perigee, 51
argument of, 53
Period. 51. 54,04
Periodic table, 228
Periodicals, io astronautics, 8
list of, 296
Permeability, 131
Permittivity, 1 10
Phase veloeitv. 100
Photoelectric effect, 172, 266
Pholooleotron, 172,242,286
Photons, 172, 173, 179, ITS, 180,211
absorption of. 205
Photosphere, 1 1
Physical const ants, 304
Pinch effect, 270
312
Index
Planck, M.. 171
I'liinik's constant. 17'. 173. I7(i
182
Planck's law, 170
Planets, flight between, 68
life on, 18
physical data for, 10
limes to reach, 70
Plasma, 275
Plasma propulsion, 35
Plato, 07
Poinenre, II., 07, 139, 191
Pollard, !■:.. 273
Positron, absorption of. 207
Positron electron pair, 243, 205
Potential, electric, 122
Potential energy, gravitational, II.
47
Potential well, in gravitational Belda,
47
for hydrogen atom, 210
Pouudal. 21
Poyntnuj'a vector, 166
Pressure of radial ion, i:t
Pressure thrust, 24
Price, 1). J„ 2!iti
Probability, wave. 8I)
Product vector, 66
Project Hover, 32
Project Sherwood, 34
Projectile motion, 43
I'ropellants. performance of, 28
Propulsion, ion, 34, 3(1
nuclear, 31
:,
solar, 36
Proton, UK, 127
Protonneutron diagram, 243, >>(*
front's hypothesis, 153
Q value. 243, 25.S
Quantization, energy, 208, "242
momentum, 209, 220
apace, 224
Quantum, 172, 173, 170
Quantum mechanics, 1X4. 2 Hi. 219
Quantum numbers, 224. 227
Quantum theory, iu7
Rad, 268
liailialioa. atmospheric contamina
tion by, 269
liiologieai effects of, 268
blnckhody, 108
cavity, 169
electromagnetic theory of, 156
101, 167
in space, 14
types of, 20 I
Radiation holts, 14
Radiation dose, 267
Radiation pressure, 13
Hadiai ion tolerances, 17
Radioactive equuibn 250
Radioactive scries, 240
Radioactivity, decav lav, for, 239
natural. 238. 249
series in, 210
lypes of, 239
units of, 24 I
Radioisotopes, natural, 251
Haniu, S., 74
ISAM) Corporation, 5, 7
Range, of charged particles, 266
of rockets. 2! 1. 30. 40, 201
for atmospheric entry, 73
Rationalised units, 1 10
Hay. 93, !!4
RayleighJeans Jaw, 17(1
Reaction principle', 20, 22
Reaction thrust, 2so
Reactor, nuclear. 273, 274
for rocket power. 32
Reduced mass, 213
Reentry of earth's atmosphere 72
Reference systems, 185
Relativistic Doppler effect, 114
Relativity theory, Einstein's, I 111
ma.vf ami energv in, IVx
Newton's, 185
and spaee travel, 201
I win paradox in, I!I5
velocity addition in, 103
Rep, 2tl.s
Resistance, of free space, 165
of metals, 147
Resistivity, 147
Resnick, *R.. 2116
Resonance, 1 10
HiKhthiiiid rule, 134
R MS speed. 83
Roberta, Michael, 1K3
Hocket, burnout velocity for, 20
definition of, 20
flight theory of, 2 1 , 20
forces on, 29
ion, 34
mass ratio, 30
model, manufacturers of, 37
motors for. 27. 32
multiple stage, 31
oosale, 28, 33
nuclear, 31
performance of, 2ii. 36
plasma, 35
propulsion of, 21, 22, 26, 32
range of. 2IJ, 30
specific impulse of, 25, 27
staging of, 31
Rocket guidance, 70
Rocket trajectories, 60
Roentgen, w, C, 77
Roentgen, 267
Rogers, H. M., 64, 206
Holler, I). H. I)., 295
Hollett, J. S., 304
Rootin can square speed (rtns), 83
Rouse, [..,!.. 296
Rowland, H. A., 2t)(i
Russell. Hortrand, 183
Rutherford atom model, 205, 238,
2S3
Ryilbcrg constant, 207
Sarnoff. !>.. 149
Satellite orbits, 47
Satellites, earth. 3
energy of, 55
escape speed for, 31
reasons for, 0, 50
Saturn, 10, 12
.Savage, J. N„ 20
Scattering of alpha particles, 204
Schoolcy, J. S., 48
Sehrodingcr, [■',., 216
Sehrodiiigcr wave equation, 222,
220, 245, 203
Scintillation counter, 262
Screening, electron. 220
Scebeek, T., 277
Sciferl, B. §., 8,88, 58
Seismic wave, 103
Semiconductor, 237, 277
Series, radioactive, 240
Shaw, J. H., 10
Shells, electron. 227, 230
Shock wave, 102
Simultaneity, 103
Singer, ('Italics, 13N
Slide rule, use of, 298
Slug, 21
Societies, professional. 296
Solar const ii at. 1 1
Solar propulsion, 30
Solar sail, 36
Solar system. 10. 18
Soli. I stale, theory ..I'. 231
Sonic boom, 102
Sound wave. 100
speed of, 101
0e» oho Wave)
Space, environment of,
gravitational fields in. 40, 47
radiation in, 14
vehicles in, 4
Space exploration, reasons for, li.
71
Spaee quantization, 224
Space research, 74
Space travel, and relativity, 201
Index 313
Space vehicles, chronology of, 4
Spec i lie heat, 87
of metals, 147
Specific impulse. 25. 27, 32, 33
Spectrum, I0S, 2<l(»
of hydrogen. 207
xray. 230
Speed, burnout, 201
of light. Hi.!. ISO. I!I2
of molecules. S3, SO
rms, 83
Spencer, II., 67
Spin, 225. 249
Sputniks, 5, 50, 05
Stable nuclei. 255
Stages, rocket, 31
(ate also Rocket 
Standing wave, I (19, 1 1 1
Star distances, 18
State, equation of, SI
StefanBoltsmann law, 170
Stern, ()., 86
Storndei huh experiment. 224
SlMwell. 0. EL III
Stake's law, 145
Stouer. R. <:.. 153
Stoney, G. J., 1 44
Strassmann, F„ 273
Strughold, II., 7
Suninierfeld, M., 38
Sun. 10
Superposition principle, 104. 109
Surge. 9S
Sutton. G. P., 26
Symbols, mathematical, 208
Thermion ie converter, 281
Thermocouple, 278
Thermoelectricity. 277
power from, 279
Thomas, B, It., !K3
Thompson, II. !.., 30(1
Thomson, J, J., 77, 131, 130, 149,
203. 288
Tli res hold energy, 25s
Thrust, decrease with alt it talc, 24
momentum, 24
pressure. 24
rocket, 24, 289
Time dilatation. 114. 195. 201
Time to reach planets. 711
Tiros satellite. 58
Total energy of B parliele, 2(10
Transformations, Lorentz, 192
Transurauic elements, 273
Trigonometric functions, 301
Tunnel effect . 245
Twin paradox, 105
Tyndall, J., 116
314
Index
I'liceifainty principle. ISI
l.'nifoniily accelerated motion, 22,
42
("nils, electrostatic, 119
tuks, 119, 120
for Newton's second law, 21
rationalized mks, I li)
Utmma (film), 18
Drams, 10, 12
\'2 rocket, 24, 30
\an Allen belts, 14, 16
Vector cross product, 50
Vector product, !2,s
Velocities, iiddition of, 193
Velocity, burnout, 29
of escape, 45
group, 1 1 in
of molecules, H'A, so
phase, Hit)
rms, 83
Venus, JO, II
Volt, 122
Von Goethe, J. W., 115
Wave, acoustic, 101, 108
at a boundary, 04, 110, 245
damped, 106
in different mediums, 94
elastic, 93
electromagnetic, 93, 103, 101
energy of, 105
group velocity in, 100
intensity of, 105
in a liquid, 07
longitudinal, 92, 103
panicle motion in. 99
phase velocity in, 100
sawtool li, 90
seismic, 103
shock, 102
sine, 96
Wave, sound, 100
speed of, 104
square, 96
standing 109, III
in u string, 95, 1 11
surge, !IS
transverse, 92, 103
traveling, 95
xray, 17.5
Wave aquation, 92, no
Schroilinger. 293
Wave forms, 95, 90, 104
Wave front, 93. :it
Wave mechanics, 219, 245, 293
\\.nii;iniile duality, 179, 217
W ave speed, !(ll
Waves, interference of, Km
soperposi'i f. KM, 109
Weber, R. I.„ 7, 153,290
Weiicr, 129
Weightlessness, \i\
Wells. II. Q 00
Whipple, V. L_ 3
White, M. \V., 290
Whitehead, A. \., 50
Whitney, W. \i., 20
Whole number rule. 153
WiedcnianuFrani! rule, 233
\\ nil's law. 171
Wilson, V. C, 281
Wilson cloud chamber, 262
Xray spectrum, characteristic, 230
continuous, 230
X rays, 175
frequency limit of, 176
scattering of, J 76
ffin, II., 19
Zodiac, 9
Zwicky, h\, 08
PHYSICAL DATA, EARTH
Mean diameter 12,742.46 km
Angular velocity 72.9 X 10~ 6 radian/sec
Mass 5.975X10" kg
Mean density 5,517 gm/cm 3
Normal gravity (p = geodetic latitude)
g m 9.78049(1 + 5.22884 X 10 3 sin 3 ^  5.9 X lO" 6 sin 2 2<p)
m/sec 2
Standard atmosphere p = 1.013 X 10 B newton/m 2
PHYSICAL CONSTANTS
Na = Avogadro's number = 6.0249 X 10 M molecules/kmole
R = Gas constant per mole = 8317 joules/ (kmole)(°K)
k  Boltzmann constant m 1.3804 X 1Q~ 23 jou1e/°K = 8.617 X
10 5 ev/°K
y Q = Standard volume of a perfect gas = 22.420 m 3 afm/kmole
c  speed of light  2.9979 X 10 s m/sec
e m Electronic charge m 1.6021 X 10~ 19 coul
h » Planck's constant = 6.6252 X 10~ 34 joulesec
F = Faraday constant « 9.652 X 10 T coul/kmole
Energy conversions: 1 electron volt = 1.6021 X 10 19 joule
1 atomic mass unit = 931.14 Mev
m c = Rest mass of electron  9.1083 X 10~ 3I kg = 0.51098 Mev
m p = Rest mass of proton = 1.6724 X 10"" kg  938.21 Mev
Mb  Rest mass of neutron m 1.6747 X 10" kg  939.51 Mev
mjm s = 1836.12
e = — c 1 = Permittivity of free space = 8.8542 X 10" 12
farad/m
^o = Permeability of free space = 4tt X 10" 7 henry/m
Zo = (/*oAo)i = Impedance of free space = 376.73 ohm
G = Universal gravitational constant = 6.67 X 10" n
newtonmVkg 2
:  : = : = =: ■"■■■■•■
* ■*
a f*
3 fii
• T
1 (0
S
68806